My project uses some libraries, which are licensed by BSD.
Can I put my project under zlib license or do I have to follow BSD and put my project under BSD as well?
The BSD licenses1 are permissive licenses. The terms of the license apply to the code that is explicitly put under the license, but the BSD licenses do not try to put any restrictions on code that is not explicitly licensed with that license.
This means that if you are using a library under a BSD license in your project, then the BSD license does not try to influence how you can license your own code. Thus, you can choose any license you like for your own code.
1: There are multiple different licenses that are referred to as BSD license. This answer applies to all of them.