An old problem in topology is to find interesting and powerful invariants of knots and 3-manifolds. Examples of such are numerical invariants, polynomial invariants or homology theory. In this talk I will explain the basic construction of some of them and outline, based on these examples, the relatively new technique of "categorification". This sophisticated theory tries to construct finer and more powerful invariants by "upgrading" old invariants using tools from different areas like representation theory, algebra, algebraic or symplectic geometry etc. In this talk I will try to illustrate with a few examples the main principle.