The P versus NP question distinguished itself as the central question of theoretical computer science nearly four decades ago. The question to resolve it, and more generally, to understand the power and limits of efficient computation, has led to the development of computational complexity theory. While this mathematical discipline in general, and the P vs. NP problem in particular, have gained prominence within the mathematics community in the past decade, it is still largely viewed as a problemof computer science.In this paper I shall try to explain why this problem, and describe the underlying concepts and problems, the attempts to understand and solve them, and some of the research directions this led us to. I shall also explain some of the important results, as well as the major goals and conjectures which stilelude us.