<p>If you have a project, that releases a library and an application, how you handle version-numbers between the two.</p>

<p>Example: Your project delivers a library, that convert different file-formats into each other. The library is released for inclusion into other applications. But you also release a command-line-application, that uses this library and implements an interface to the functionality.</p>

<p>New releases of the library lead to new releases of the application (to make use of all new features), but new releases of the application may not trigger new releases of the library. Now how are the versions numbers handled: Completely independent or should library- and application-version be dependent in some way?</p>

Completely independent version numbers, but the command line (or any other dependent) app should say which version of the library it was compiled against in the help section or a banner.
That way you will be able to tell which functionality will the apps have and reduce potential confusion, especially given that somebody could compile a newer app version against an old library for any reason. Also, you decouple them and can add features on the library without depending on release of a new app version and so on.
If you are sure you will always want all the apps and library to go in lockstep then you could use same numbers, but that's adding a constraint for not a strong reason.