Abstract The first aim of this work is to introduce a definition and properties of the derived functors Ext and Tor. We then consider the notions of homological dimensions of modules; namely, projective, injective and flat dimensions, for which we present characterization using the functors Ext and Tor. We also present the homological dimensions of rings; namely, global and weak dimensions, in the context of characterizing semisimple , Dedekind, Von Neumann, and Prufer rings. We finally study the transfer of global and weak dimensions over polynomial rings.