OpenSSL 3.0.0-alpha1 was releases about a week ago. OpenSSL 1.1.x APIs are still functional. However some APIs have been deprecated and FIPS (usedforsecurity flag) is no longer functional.

* One shot HMAC() is deprecated and should be replaced with EVP_MAC API calls
* ERR_func_error_string() is deprecated
* OpenSSL has introduced a new concept of crypto providers (OSSL_PROVIDER), library context (OPENSSL_CTX) and additional flags. A new function EVP_MD_fetch() has been introduced.
* FIPS support has been rewritten and is now shipped with OpenSSL 3.0.0. EVP_MD_CTX_FLAG_NON_FIPS_ALLOW is no longer supported. FIPS state is no longer part of EVP_MD_CTX but of EVP_MD.

See https://wiki.openssl.org/index.php/OpenSSL_3.0

New changeset 16d4e6f6f559b4fd21c9d29fea303489f658674f by Christian Heimes in branch 'master':
bpo-40479: Fix hashlib issue with OpenSSL 3.0.0 (GH-20107)
https://github.com/python/cpython/commit/16d4e6f6f559b4fd21c9d29fea303489f658674f

New changeset 62d618c06bd395308b7163dbcb26c7e6d0922033 by Christian Heimes in branch 'master':
bpo-40479: Test with latest OpenSSL versions (GH-20108)
https://github.com/python/cpython/commit/62d618c06bd395308b7163dbcb26c7e6d0922033

New changeset 5e6b491403d7211588dcd399167f5bc21781c69c by Miss Islington (bot) in branch '3.7':
bpo-40479: Test with latest OpenSSL versions (GH-20108)
https://github.com/python/cpython/commit/5e6b491403d7211588dcd399167f5bc21781c69c

New changeset 5a06cf01ecb6a048fb47c086adc1336f54fe8789 by Miss Islington (bot) in branch '3.8':
bpo-40479: Test with latest OpenSSL versions (GH-20108)
https://github.com/python/cpython/commit/5a06cf01ecb6a048fb47c086adc1336f54fe8789

Simple benchmark with openssl-3.0.0-5.el9.x86_64

With custom EVP_MD cache:
$ ./python -m timeit -s "from hashlib import md5" "md5(b'12345678', usedforsecurity=False).digest()"500000 loops, best of 5: 520 nsec per loop
$ ./python -m timeit -s "from hashlib import sha512" "sha512(b'12345678', usedforsecurity=False).digest()"
500000 loops, best of 5: 730 nsec per loop

Without EVP_MD cache:
$ ./python -m timeit -s "from hashlib import md5" "md5(b'', usedforsecurity=False).digest()"
500000 loops, best of 5: 807 nsec per loop
$ ./python -m timeit -s "from hashlib import sha512" "sha512(b'12345678', usedforsecurity=False).digest()"
200000 loops, best of 5: 1.03 usec per loop

New changeset 443b308fee088e21bbf472c376c5c9e3648f916c by Christian Heimes in branch 'main':
bpo-40479: Fix hashlib's usedforsecurity for OpenSSL 3.0.0 (GH-30455)
https://github.com/python/cpython/commit/443b308fee088e21bbf472c376c5c9e3648f916c

Something went wrong with Miss Islington. 
3ce6945f5f434806eea700eb5ff1bed6d39395e1 / GH-30573 is backport to 3.10.

New changeset 4ddd5da2691bea39e36debbc7f53c7cc4f13904e by Christian Heimes in branch '3.9':
[3.9] bpo-40479: Fix hashlib's usedforsecurity for OpenSSL 3.0.0 (GH-30455) (GH-30574)
https://github.com/python/cpython/commit/4ddd5da2691bea39e36debbc7f53c7cc4f13904e

New changeset 276c234ce0fa6732237f1b187989837324d9dea3 by Christian Heimes in branch 'main':
bpo-40479: Fix typo, flag must be set for OpenSSL < 3.0.0 (GH-30584)
https://github.com/python/cpython/commit/276c234ce0fa6732237f1b187989837324d9dea3

New changeset 47422a852de14a8ec11d058136c7c864d2cc7fc9 by Miss Islington (bot) in branch '3.10':
[3.10] bpo-40479: Fix typo, flag must be set for OpenSSL < 3.0.0 (GH-30584) (GH-30585)
https://github.com/python/cpython/commit/47422a852de14a8ec11d058136c7c864d2cc7fc9

New changeset 537f16adfa31b5b1fe9d656d571d1e10fb115351 by Miss Islington (bot) in branch '3.9':
bpo-40479: Fix typo, flag must be set for OpenSSL < 3.0.0 (GH-30584)
https://github.com/python/cpython/commit/537f16adfa31b5b1fe9d656d571d1e10fb115351

There is a missing call to va_end() in 443b308fee088e21bbf472c376c5c9e3648f916c (which is undefined behavior).  PR 31153 fixes this.

New changeset 59e004af63742361b67d1e1ae70229ff0db1059d by Zackery Spytz in branch 'main':
bpo-40479: Fix undefined behavior in Modules/_hashopenssl.c (GH-31153)
https://github.com/python/cpython/commit/59e004af63742361b67d1e1ae70229ff0db1059d

New changeset 3ceff9968be059209f0d7271365f38895207b24d by Miss Islington (bot) in branch '3.10':
bpo-40479: Fix undefined behavior in Modules/_hashopenssl.c (GH-31153)
https://github.com/python/cpython/commit/3ceff9968be059209f0d7271365f38895207b24d

New changeset 0892a0e3ec925ac959bc959b985293b4b96ef863 by Miss Islington (bot) in branch '3.9':
bpo-40479: Fix undefined behavior in Modules/_hashopenssl.c (GH-31153)
https://github.com/python/cpython/commit/0892a0e3ec925ac959bc959b985293b4b96ef863