I've been reading Coleman's notes and the book on QFT by Ticciati. There they both place a lot of emphasis on computing the scattering matrix S. I can follow their computations (using Wick's theorem etc.) but I don't really have a good understanding of what S actually tells you. Ticciati even mentions that once you have S you've solved the theory. Why is this the case? In order to get another viewpoint I started reading the QFT book by Peskin & Schroeder. Here they start with the computation of the correlation functions. How are these n-point correlation functions related to S? Thanks in advance.