The analysis of distribution and statistical independence properties of some classical and recent methods for the generation of uniform pseudorandom numbers leads to interesting types of character sums. Some of these character sums are well known, like Kloosterman sums, while others are new types. We report on bounds for such character sums and their implications for the underlying pseudorandom numbers. Special emphasis will be placed on new methods that are required to treat certain of these character sums, in particular in the case of incomplete character sums. These new techniques are of general interest for the theory of character sums.