Prime numbers have fascinated mathematicians, both professional and amateur, for centuries. They are often called the "building blocks" of the set of positive integers because every positive integer either is a prime number itself or can be written as the product of prime numbers in exactly one way. Few objects in mathematics are as simple to describe, yet few have so much depth and mystery to their structure. These numbers (2, 3, 5, 7, 11, ...) show up seemingly at random, yet some beautiful patterns underlie their distribution. This chapter explores some of these patterns and connects the prime numbers to some of the deepest problems in all of mathematics.