What's the level cap in Skyrim?

In Oblivion, the way the game worked, the max level possible was based solely on your choices for primary skills, and could vary anywhere from between 45 and 53 (with soft-caps at 29 and 35 in terms of NPC leveling) or when you maxed all of your character attributes at 100.

What's the level cap in Skyrim? Is it a hard or soft cap -- once you hit it, can you continue to 'level' your skills and unlock perks anyway?

Since patch 1.9 there is no maximum level anymore.

If you have a skill at 100 you can make it "Legendary". This resets the skill to 15, gives all perk points invested in that specific tree back. Be warned: You lose all the benefits of having a higher skill level! The advantages of making skills legendary are:

• Difficulty increase, since the enemies don't have their skills reduced.
• You can continue to level up and earn extra skill perks.

It is possible to repeatedly make a skill legendary or to make multiple skills legendary, this allows you to (slowly) unlock all perk points.

Level 81 is the maximum level in Skyrim without making any skill Legendary

There is a linearly constant amount of "experience" required for each level. The formula is simply:

(level-1) * 25 + 100

Where level is your current level.

"Experience" in this case is earned strictly through skill ups. Leveling a skill to X will give you X experience towards your next level. For instance, to level from level 1 to level 2, you require 100 experience. This means you can either raise 5 skills to level 20, one skill to level 21 (assuming it starts at 15), or any other permutation thereof.

Also, and this is important:

There is no soft cap

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Level 81 is just as (relatively) easy to reach as Level 11, it just requires more total experience.

Since all races have +10 to one skill and +5 to five skills, this means there is a total of 88,085 "experience" to earn, and this is constant across the ten races. Level 82 would require 89,100 total experience, and is currently unreachable.

Thus, when your character maxes all of their skills, they will be level 81.

According to Todd Howard, game director at Bethesda, there is a mathematical cap somewhere in the 70s. Although most players will max out at level 50.

See this article for details.

Since you gain perk points each time you level up, and levels only by increasing your skills, the maximum level is when all of your skills are lvl 100.

For my cat I tested this with (using console commands) I managed to reach lvl 81 (and a bit), for a total of 80 perks.

As of Patch 1.9, the level cap of 81 has been removed.

^{(This is an update to the current answers, regarding Patch 1.9's changes.)}
At higher levels, leveling up happens much more slowly. Level increases in Skyrim follow a formula (detailed here). Prior to Patch 1.9, the maximum level was 81, since there was no way to gain any more experience once every skill reached its maximum value of 100.

With Patch 1.9, individual skills can be made "Legendary", denoting them with an Imperial symbol. This will reset the skill to 15, and perks used for that skill may be redistributed. Gaining levels in that skill will affect leveling again, thus effectively removing the level cap of 81. There is no restriction on the number of times that skills can be made Legendary. This change makes it theoretically possible to obtain every single perk in the game.

According to the formulae, training one individual skill all the way from 15 to 100 (one mastering cycle) yields 4,930 XP. This amount of XP is enough to level from 1 to 17, or from 194 to 195. Leveling up to 252 (unlocking all perks) would take a total of 809,475 XP. This corresponds to over 164 skill mastering cycles, or mastering every skill over 9 times. The proper approach to such a huge task would be to find the easiest skill to raise and to exclusively focus on that for leveling purposes.

^{Source: UESP wiki, "Skyrim - Leveling" article}