In this paper, we describe and compare two approaches to reasoning about nonblocking algorithms. We give a brief overview of the simulation approach we have used in previous work. We then give a more detailed description of an approach based on Lipton's reduction method, and illustrate it by verifying two versions of a shared counter and two versions of a shared stack. Both approaches work by transforming a concurrent execution into an equivalent sequentia-execution, but they differ in the way that executions are transformed and the way that transformations are justified.