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First, let's establish that we often refer to TLS and SSL interchangeably as SSL here. The current protocol is called TLS, it was called SSL a long time ago.
There are several known reasons why a connection that involves SSL might fail. This is a document that attempts to details the most common ones and how to mitigate them.
CA certs are used to digitally verify the server's certificate. You need a "ca bundle" for this. See lots of more details on this in the SSLCERTS document.
When using said CA bundle to verify a server cert, you will experience problems if your CA cert does not have the certificates for the intermediates in the whole trust chain.
Some broken servers fail to support the protocol negotiation properly that SSL servers are supposed to handle. This may cause the connection to fail completely. Sometimes you may need to explicitly select a SSL version to use when connecting to make the connection succeed.
An additional complication can be that modern SSL libraries sometimes are built with support for older SSL and TLS versions disabled!
All versions of SSL are considered insecure and should be avoided. Use TLS.
Clients give servers a list of ciphers to select from. If the list doesn't include any ciphers the server wants/can use, the connection handshake fails.
curl has recently disabled the user of a whole bunch of seriously insecure ciphers from its default set (slightly depending on SSL backend in use).
You may have to explicitly provide an alternative list of ciphers for curl to use to allow the server to use a WEAK cipher for you.
Note that these weak ciphers are identified as flawed. For example, this includes symmetric ciphers with less than 128 bit keys and RC4.
WinSSL in Windows XP is not able to connect to servers that no longer support the legacy handshakes and algorithms used by those versions, so we advice against building curl to use WinSSL on really old Windows versions.
BEAST is the name of a TLS 1.0 attack that surfaced 2011. When adding means to mitigate this attack, it turned out that some broken servers out there in the wild didn't work properly with the BEAST mitigation in place.
To make such broken servers work, the –ssl-allow-beast option was introduced. Exactly as it sounds, it re-introduces the BEAST vulnerability but on the other hand it allows curl to connect to that kind of strange servers.
Some SSL backends may do certificate revocation checks (CRL, OCSP, etc) depending on the OS or build configuration. The –ssl-no-revoke option was introduced in 7.44.0 to disable revocation checking but currently is only supported for WinSSL (the native Windows SSL library), with an exception in the case of Windows' Untrusted Publishers blacklist which it seems can't be bypassed. This option may have broader support to accommodate other SSL backends in the future.