This paper is a comprehensive review of the research conducted over the past four decades in the domain of time-dependent scheduling, where variable processing times of jobs depend on when the jobs start. The paper is divided into four parts. The first part recalls some definitions and notions, introduces terminology, and defines the main models of time-dependent job processing times and the notation that is used throughout the paper. The second part summarizes four decades of time-dependent scheduling research, focusing on the computational complexity of time-dependent scheduling problems, and algorithms solving these problems. The results are divided into groups with respect to the machine environment and illustrated by examples. The third part concentrates on new topics in time-dependent scheduling, such as two-agent time-dependent scheduling, mutually related time-dependent scheduling problems, and time-dependent scheduling games. The last part discusses the most important time-dependent scheduling problems which still await solution. The paper is completed by bibliographic notes and an extensive list of references.