Crack determination problems refer to a class of mathematical problems formulated as idealisations of the engineering practices of detecting and determining cracks in structural members. These problems are classified as inverse problems of determining unknown domains and boundaries. This paper reviews efforts made so far, and presents some new results in mathematical, numerical and experimental studies of crack determination problems. We shall be mainly concerned with two typical special cases, namely, the Laplace and elastodynamic cases. The crack determination problem for Laplace's equation is quite remarkable in that it attracted both engineers and mathematicians. Indeed, we can find an example of fruitful interactions between science and technology in this particular problem. This made it possible to include mathematical results such as uniqueness etc. in this paper, in addition to numerical techniques to solve the problem. The elastodynamic case, on the other hand, is closely related to engineering practices of nondestructive evaluation using ultrasound, and has so far been investigated mainly by engineers from the view point of numerical analysis. In this paper we shall present numerical techniques for this problem, and see how one can extend numerical techniques developed for the Laplace cases to problems in time domain. This paper concludes with some comments on other examples of crack determination problems.