The paper gives a clear and interesting review on the main problems, concepts and results in the area of information-based complexity, a field receives growing interest in computer science and computational mathematics. The basic problem can be formulated as follows: Given a class F of problems, a solution operator \(S: F\to G\), G a normed linear space, an approximation x(f) to S(f) is obtained by (1) selecting some information N(f) from the given problem and (2) applying a finite algorithm \(\phi\) : N(F)\(\to F\) to obtain x(f). The question is, what can be said about the complexity of computing x(f) such that \(\| x(f)- S(f)\| <\epsilon\), \(\epsilon\) a given tolerance. The worst as well as the average case are discussed and for linear problems a number of results and open questions are stated.