In this chapter we explain how to analyze the time an algorithm takes to solve a given problem and specifically whether it takes polynomial or exponential time. We consider a variety of well-known problems from computer science to illustrate polynomial-time versus exponential-time algorithms. In these analyses, we build on a common distinction between three types of problems: optimization problems, search problems, and decision problems. While the first two are most commonly adopted in cognitive science, the last one is widely used for computational complexity analyses. As we will explain in subsequent chapters, the distinction is overcome during complexity analyses by understanding the close relationship between these problem types. We will also see that while it is possible to "prove by example" that a problem is of polynomial-time complexity (viz., give an algorithm that solves the problem that runs in polynomial time) proving that a problem does not allow for any such polynomial-time algorithm requires different proof methods.