While it is true that the class P is the set of all problems which have an algorithmic solution in polynomial time by a deterministic Turing machine, the class NP is the set of all problems which have an algorithmic solution in polynomial time by a non-deterministic Turing machine.
There are problems which can be solved in exponential time which are not in NP; NP is a proper subset of exponentially solvable problems.
An equivalent description of the class NP is the set of problems whose answers (once gotten from some magic oracle or God or whatever your ontological committments demand) can be verified in polynomial time by a deterministic Turing machine. The Travelling Salesman Problem can be verified in polynomial time (hand me the itinerary, and I can say either "Yes" or "No" very quickly to the question of whether it visits each vertex on a non-complete graph without repetition) by a deterministic Turing machine, but no algorithm has yet been found to solve this problem in polynomial time on a deterministic Turing machine.