Testing if a multivariate polynomial given as an arithmetic circuit is identically zero is a fundamental problem in the theory of computation. It has been studied by computer scientists and mathematicians for about thirty years. From early on, there have been efficient randomized algorithms solving the problem. However, designing efficient algorithms that use fewer or no random bits at all has turned into a notorious open problem over the years. By now, it is understood that a deterministic algorithm for general arithmetic circuits would have major consequences in theoretical computer science. To approach this goal, it is worthwhile to understand the randomness complexity of polynomial identity testing in restricted models. In this book, we consider some natural and well-studied models in which we obtain new results.