Is it "despite" or "despite of"?

Should I always use 'despite' instead of 'despite of'?

As JSBangs and Kosmonaut have pointed out already, despite is the way to go in contemporary English.

However, despite of is not incorrect per se; it's just a bit dated. Look no further than at the works of William Shakespeare:

• "Grace is grace, despite of all controversy: as, for example, thou thyself art a wicked villain, despite of all grace." (Measure for Measure).
• "The scar that will, despite of cure, remain" (Rape of Lucrece).
• "Some good I mean to do, / Despite of mine own nature" (King Lear)
• "but my fair name, / Despite of death that lives upon my grave, / To dark dishonour's use thou shalt not have." (Richard II).
• "So thou through windows of thine age shall see / Despite of wrinkles this thy golden time." (Sonnet III).
• "Therefore, despite of fruitless chastity, / Love-lacking vestals and self-loving nuns, / That on the earth would breed a scarcity / And barren dearth of daughters and of sons, / Be prodigal" (Venus and Adonis).
• "For then despite of space I would be brought, / From limits far remote where thou dost stay." (Sonnet XLIV)
• ...

What's more, that quirky fella didn't stop at that. Just look at this horrible English:

• "O, a good wish upon you! You will try in time, in despite of a fall." (As You Like It)
• "But 'tis my heart that loves what they despise, / Who in despite of view is pleased to dote;" (Sonnet CXLI)
• "No, in despite of sense and secrecy, / Unpeg the basket on the house's top, / Let the birds fly" (Hamlet)
• "How France and Frenchmen might be kept in awe, / And had his highness in his infancy / Crowned in Paris in despite of foes?" (Henry VI, Part II)
• "Deposed he shall be, in despite of all." (Henry VI, Part III)
• "Yet, Warwick, in despite of all mischance, / Of thee thyself and all thy complices, / Edward will always bear himself as king" (Henry VI, Part III)
• "Then, in despite of brooded watchful day, / I would into thy bosom pour my thoughts" (King John)

I (well, he) could go on, but you get the idea.

Now let's head over to the States and look at what the Corpus of Historical American English has to offer:

DESPITE OF        272
IN DESPITE OF     169

That's not much, but not exactly nothing, either. The most interesting part is the distribution by year. A picture is worth a thousand words:

_{(X axis: year, Y axis: incidences per million words)}

Now, before you forget where I started, I will add that the Corpus also has a whopping 24305 cites for just "despite", and 20802 cites for "in spite of". So a more accurate picture would be this:

Note how this also tells an interesting story about despite vs. in spite of.

Back to the UK, the British National Corpus has 2695 cites for "in spite of", 14356 cites for "despite", but only 3 cites for "(in) despite of".

So the bottom line is this: by all means do use despite (or in spite of). However, if you accidentally use despite of or even in despite of and someone corrects you, you can now at least pretend that you were emulating Shakespeare.

Edit: Interestingly enough, Merriam-Webster simply says that in despite of is a synonym for in spite of and leaves it at that. I also totally forgot to look it up in the most obvious place: Etymonline. Here's what it has to say:

despite [...] The preposition (early 15c.) is short for in despite of (late 13c.), a loan-translation of Old French en despit de "in contempt of."

Yes, you should use despite. The word despite is a preposition which takes a noun as its object, and doesn't require of. Despite of is incorrect, and sounds distinctly non-native.

(You're probably getting confused by the similar phrase in spite of. In spite of means basically the same thing as despite.)

When you say "despite of", it sounds like you are conflating "despite" and "in spite of". "Despite of" is not a phrase, but these other two are. Their meanings are basically the same.

'Despite' as a preposition, a noun, and a verb

The three answers from 2010 pay little or no attention to the fact that despite can function as either a preposition or a noun—although RegDwigнt does cite Etymonline as observing that "The preposition [despite] (early 15c.) is short for in despite of (late 13c.)" In the phrase "in despite of," despite is a noun.

In Merriam-Webster's Eleventh Collegiate Dictionary (2003), the entry for despite as a preposition is quite brief:

despite prep (15c) : in spite of {played despite an injury}

The entry for despite as a noun is much longer and more interesting:

despite n {ME, fr. AF despit, fr. L despectus, fr. despicere} (13c) 1 : the feeling or attitude of despising : CONTEMPT 2 : MALICE, SPITE 3 a : an act showing contempt or defiance b : DETRIMENT, DISADVANTAGE {I know of no government which stands to its obligations, even to its own despite, more solidly —Sir Winston Churchill}

The quotation from Churchill in the Eleventh Collegiate's entry for despite as a noun appears in the context of a comment by Churchill about the Soviet Union, seemingly delivered soon after the famous 1945 Yalta conference involving Churchill, Stalin, and Roosevelt:

The impression I brought back from the Crimea, and from all my other contacts, is that Marshal Stalin and the Soviet Leaders wish to live in honorable friendship and equality with the Western democracies. is that western democracies. I feel also that their word is their bond. I know of no government which stands to its obligations even to its own despite, more solidly than the Russian Soviet Government.

Anyway, the striking thing about the Eleventh Collegiate's coverage of despite as a noun is that it doesn't include any labels indicating that despite is little used as a noun in modern English. Contrast that with the way the same dictionary handles the entry for despite as a verb:

despite vt despited, despiting (14c) 1 archaic : to treat with contempt 2 : obs : to provoke to anger : VEX

From this difference in treatment between the verb despite and the noun despite, it seems clear that Merriam-Webster considers despite as a noun to be neither archaic nor obsolete, as of 2003.

Merriam-Webster had some grounds for reaching this conclusion. Less than a century before, The Century Dictionary and Cyclopedia (1906) had taken the position that despite as a preposition was merely a short form of the phrase "in despite of," meaning "notwithstanding." The Century Dictionary devotes this paragraph to "in despite of" under its entry for despite as a noun:

In despite of, in defiance or contempt of: in defiant opposition to: notwithstanding: later abbreviated to in spite of, or simply despite as a preposition.

Notably, the Century Dictionary doesn't say that "in despite of" can correctly be shortened to "despite of." The reason it doesn't, I think, is that without the leading preposition in, the despite in "despite of" becomes a preposition—and when despite functions as a preposition, it normally doesn't take in before it or of after it.

'Despite of' and 'in despite of' in Google Books search results

A search for the phrases "despite of" (blue line), "in despite of" (red line) and "in his despite" (green line) across the period 1650–2008 shows the extent to which matches for "despite of" are also matches for "in despite of":

As the chart indicates, until about 1825, the match between the two forms is such that vast majority of instances of "despite of" are also matches for "in despite of"; instances of "despite of" without in immediately preceding it are sufficiently rare that the blue line emerges as a kind of blue snow cap on some of the summits of a red mountain range. If we move the start of the chart to 1687, to avoid the spike from 1684 that dwarfs the rest of the chart, we get this view of the Google Books results for 1687–2008:

The snowcapping phenomenon is evident until about 1825, when searches find enough matches for "despite of" that aren't also matches for "in despite of" to establish significant separation between the two terms' line graphs. The individual Google Books search results for "despite of" for the years 1827–1837 continue to show most matches as involving instances of "in despite of"—but they also show some that don't—including this example from Letter XIII of Walter Scott, Paul's Letters to His Kinfolk (1816), reproduced in The Miscellaneous Prose Works of Sir Walter Scott (1827, 1829, 1833, 1834):

But Napoleon knew well the people over whom he was called to rule, and was aware that his power was secure, despite of annihilated commerce and exhausted finances, despite of his waste of the lives of Frenchmen and treasure of France, despite of the general execration of the human race, echoed from the Baltic to the Mediterranean, providing he could prove to the Parisians that he was still the Emperor of the World, and Paris its capital.

Still, instances of "despite of" without a leading in remained less common than the form "in despite of"—and of course far less common than the simple preposition despite by itself—at least through the end of the nineteenth century.

'Despite of' today

Cambridge Advanced Learner's Dictionary (2008) offers this advice on the phrase "despite of":

Common mistake: despite

Remember that despite is never followed by 'of'. Don't say 'despite of something', say despite something:

She is an excellent assistant, despite of her lack of computer skills.

She is an excellent assistant, despite her lack of computer skills.

Note that in spite has a similar meaning but is always followed by 'of':

In spite of her lack of computer skills, she is an excellent assistant.

The entry for despite as a preposition in this Cambridge dictionary consists of two definitions:

despite preposition 1 without taking any notice of or being influenced by; not prevented by: [examples omitted] 2 despite yourself If you do something despite yourself, you do it although you do not want to or although you know you should not: [examples omitted]

This dictionary, however, has no entry for despite as a noun.

Recent instances of "despite of" as a prepositional phrase are somewhat rare in Google Books searches, but they do occur. For example, from Jorge Alió & ‎Dimitri Azar, Management of Complications in Refractive Surgery (2008):

Dry eye symptoms have not been shown to increase after myopic LASIK enhancements despite of documented higher ocular surface fluorescein staining scores.

And from Nathan Rein, The Chancery of God: Protestant Propaganda Against the Empire, Magdeburg 1546–1551 (2008):

Reconciliation between Magdeburg and the Empire still seemed possible, negotiations were under way, and a military campaign would be horrendously expensive; thus, Maurice and the other princes held out for a peaceful settlement, even despite of the mounting evidence of Magdeburg's intransigence in the form of pamphlets and occasional anticlerical aggression.

There are even some recent instances of "in despite of," with despite clearly operating in its old position as a noun. For example, from John France, The Crusades and the Expansion of Catholic Christendom, 1000–1714 (2006):

Strongbow had married Diarmait's daughter Aoife, and on her father's death it became known that the dead king had designated him as his successor to Leinster, in despite of Irish succession laws and the claims of his own sons.

And from Hanning Chen, Molecular Modeling of Proton Transport in Condensed Phases (2008):

Examining th hydronium coordination number of NEO as a function the mole ratio between HCl and NEO reveals that the hydrophobic surface of NEO is large enough to accommodate the amphiphilic hydroniums even for A4 as ~80% of the hydroniums participate in the formation of HHC in despite of the low pH value.

...

Nevertheless, in despite of the small numerical discrepancy, the FM-EVB is still able to correctly reflect the tendency of the boxsize effect on D_H⁺, and is also projected to be in reasonably good agreement with the FP-EVB even for considerably large systems.

Conclusions

Cambridge Advanced Learner's Dictionary is clearly incorrect when it asserts that "despite is never followed by 'of'," but the rule that the dictionary tries to teach advanced English learners is undoubtedly practical and useful: most people speaking and writing in English don't use the phrase "despite of," and many people hearing the phrase will assume that it is an error.

The facts that despite in the phrase "in despite of" is a noun and that (according to the most recent Merriam-Webster's Collegiate Dictionary) despite as a noun remains in good standing in current English hold little practical weight, it seems to me. In most real-world situations, "in despite of" comes across as poetical, quaint, or mistaken. Still, the phrase has a long history of respectable use in English, and the notion that "in despite of" somehow transgresses against sense even more objectionably than "despite of" alone does fails to consider the differing parts of speech that despite functions as in the two phrases.