A non-privileged user or program can put code and a config file in a known non-privileged path (under C:/usr/local/) that will make curl <= 7.65.1 automatically run the code (as an openssl "engine") on invocation. If that curl is invoked by a privileged user it can do anything it wants.
Curl versions 7.14.1 through 7.61.1 are vulnerable to a heap-based buffer over-read in the tool_msgs.c:voutf() function that may result in information exposure and denial of service.
curl before 7.53.0 has an incorrect TLS Certificate Status Request extension feature that asks for a fresh proof of the server's certificate's validity in the code that checks for a test success or failure. It ends up always thinking there's valid proof, even when there is none or if the server doesn't support the TLS extension in question. This could lead to users not detecting when a server's certificate goes invalid or otherwise be mislead that the server is in a better shape than it is in reality. This flaw also exists in the command line tool (--cert-status).
curl version curl 7.20.0 to and including curl 7.59.0 contains a CWE-126: Buffer Over-read vulnerability in denial of service that can result in curl can be tricked into reading data beyond the end of a heap based buffer used to store downloaded RTSP content.. This vulnerability appears to have been fixed in curl < 7.20.0 and curl >= 7.60.0.
curl before version 7.52.1 is vulnerable to an uninitialized random in libcurl's internal function that returns a good 32bit random value. Having a weak or virtually non-existent random value makes the operations that use it vulnerable.
A buffer overflow exists in curl 7.12.3 to and including curl 7.58.0 in the FTP URL handling that allows an attacker to cause a denial of service or worse.
A buffer over-read exists in curl 7.20.0 to and including curl 7.58.0 in the RTSP+RTP handling code that allows an attacker to cause a denial of service or information leakage