A sequence $(a_1,..., a_n)$ of positive integers is a parking function if its nondecreasing rearrangement $(b_1 \leq ... \leq b_n)$ satisfies $b_i < i + 1$ for all $i$. Parking functions were introduced by Konheim and Weiss to study a hashing problem in computer science, but have since received a great deal of attention in algebraic combinatorics. We will define two new objects attached to any (finite, real, irreducible) reflection group which generalize parking functions and deserve to be called parking spaces. We present a conjecture (proved in some cases) which asserts a deep relationship between these constructions. This is joint work with Drew Armstrong at the University of Miami and Vic Reiner at the University of Minnesota.