The topological complexity of a problem over the real numbers is the minimum number of tests or comparisons that one needs to perform in order to solve the problem. One can consider topological complexity in various settings. It has turned out that the topological complexity of a problem can be very different depending on the set of operations which, besides comparisons, are allowed for its solution: either only algebraic operations, or also other operations like exp, log, absolute value, or even all continous operations, which is the purely topological setting. In the purely topological setting the topological complexity can be characterized also as a degree of discontinuity. It is closely related to the Schwarz genus, and one is led to problems involving algebraic topology, investigated by Smale and Vassiliev. Computational problems in various fields give rise to an investigation in topological complexity, namely algebraic complexity theory, numerical computation, and computational geometry. For example in computational geometry the topological complexity of a problem can be considered as a measure of the degree of degeneracy of the degenerate input configurations, i.e., those configurations where the output values or some values computed during the solution process show a discontinuity. In this talk we give an introduction to topological complexity. After presenting fundamental definitions and results, we will focus on results concerning the topological complexity of various versions of the problem to approximate zeros of univariate functions on the unit interval, studied by Novak and Wozniakowski and by the speaker.