The authors give a selection of randomized algorithms in number theory. They are called randomized because some random choices are made in the course of the execution. Usually this means that the run time of the algorithms can become infinitely long. However, the expected run time is finite. Often most executions have a run time that is near the expected value. Randomized algorithms are used when the calculated expected run time, and the observed run time in practice, is shorter than that of deterministic algorithms. In this paper the authors present a selection of randomized algorithms to find representations of natural numbers as sums of two, three or four squares, or as sums of three triangular numbers. Most of these algorithms are already published at other places. This paper is meant to give an overview of what is available. The authors give expected run times for the algorithms, all of which are polynomial in the logarithm of the number to be represented. In some cases the expected run time is dependent on the truth of some unproven conjectures. In practice the given algorithms are fast. In any case the expected run time is (in the limit) faster than deterministic algorithms performing the same task.