Deterministic replay is a type of emerging technique dedicated to providing deterministic executions of computer programs in the presence of nondeterministic factors. The application scopes of deterministic replay are very broad, making it an important research topic in domains such as computer architecture, operating systems, parallel computing, distributed computing, programming languages, verification, and hardware testing. In this survey, we comprehensively review existing studies on deterministic replay by introducing a taxonomy. Basically, existing deterministic replay schemes can be classified into two categories, single-processor (SP) schemes and multiprocessor (MP) schemes. By reviewing the details of these two categories of schemes respectively, we summarize and compare how existing schemes address technical issues such as log size, record slowdown, replay slowdown, implementation cost, and probe effect, which may shed some light on future studies on deterministic replay.