Can an email address contain international (non-english) characters?

If it's possible, should I accept such emails from users and what problems to expect when I will be sending mails to such addresses?

Solution 1:

Officially, per RFC 6532 - Yes.

For a quick explanation, check out wikipedia on the subject.

Solution 2:

Update 2015: Use RFC 6532

The experimental 5335 has been Obsoleted by: 6532 and
this later has been set to "Category: Standards Track",
making it the standard.

The Section 3.2 (Syntax Extensions to RFC 5322) has updated most text fields to
include (proper) UTF-8.

The following rules extend the ABNF syntax defined in [RFC5322] and
[RFC5234] in order to allow UTF-8 content.

VCHAR   =/  UTF8-non-ascii
ctext   =/  UTF8-non-ascii
atext   =/  UTF8-non-ascii
qtext   =/  UTF8-non-ascii
text    =/  UTF8-non-ascii
             ; note that this upgrades the body to UTF-8
dtext   =/  UTF8-non-ascii

The preceding changes mean that the following constructs now
allow UTF-8:
   1.  Unstructured text, used in header fields like
       "Subject:" or "Content-description:".
   2.  Any construct that uses atoms, including but not limited
       to the local parts of addresses and Message-IDs. This
       includes addresses in the "for" clauses of "Received:"
       header fields.
   3.  Quoted strings.
   4.  Domains.

Note that header field names are not on this list; these are still
restricted to ASCII.

Please note the explicit inclusion of Domains.
And the explicit exclusion of header names.

Also Note about NFKC:

The UTF-8 NFKC normalization form SHOULD NOT be used because
it may lose information that is needed to correctly spell
some names in some unusual circumstances.

And Section 3 start:

Also note that messages in this format require the use of the
SMTPUTF8 extension [RFC6531] to be transferred via SMTP.

Solution 3:

The problem is that some mail clients (server-tools and / or desktop tools) don't support it and throw an 'invalid email' exception when you try to send a mail to an address which contains umlauts for example.

If you want full support, you could do the trick with converting the email-address parts to "punycode". This allows users to type in their addresses the usual way but you save it the supported-level way.

Example: müller.com » xn--mller-kva.com

Both points to the same thing.