Computation has become a standard tool in just about every academic field. Whole subfields of biology, chemistry, physics, economics and others are devoted to large-scale computational modeling, simulations, and problem solving.
As we solve larger and more complex problems with greater computational power and cleverer algorithms, the problems we cannot tackle begin to stand out. The theory of NP-completeness helps us understand these limitations and the P versus NP problem begins to loom large not just as an interesting theoretical question in computer science, but as a basic principle that permeates all the sciences.
Really interesting discussion of P=NP at ACM Magazine -- where are we at now, and what's going to happen to the problem?
Edit: This article was current until this: