Computational topology is a branch of computational mathematics - a new discipline emerging inbetween mathematics and computer science and aiming in obtaining mathematically rigorous results with a help of computers. Recently, computational topology gained a lot of attention in various fields outside mathematics. I will start this talk by summarizing some of those applications. Later the concept of persistent homology will be explained and the algorithm(s) to compute it for a finite filtered cell complex will be given. Having this knowledge we will consider the problem of rigorous computations of level sets (up to homotopy type) and persistent homology of sufficiently smooth functions f: R^n --> R .