In this chapter we introduce the classical complexity classes P and NP, where P consists of decision problems that are solvable in polynomial time and NP consists of decision problems for which yes-instances are verifiable in polynomial time. We show that the latter problem class contains many problems for which it is conjectured that no polynomial-time algorithm exists. We introduce the formal notions of NP-hardness and NP-completeness to show that a problem is as least as hard as the hardest problems in NP, and we describe why NP-hard problems are considered to be intractable (i.e., not computable in polynomial time). We explain how one can prove that a problem is a member of either P or NP, and how one can use the technique of polynomial-time reduction, introduced and practiced in Chapter 3, to prove NP-hardness and NP-completeness.