  Let $W$ be a Weyl group with root lattice $Q$ and Coxeter number $h$. The
elements of the finite torus $Q/(h+1)Q$ are called the $W$-{\sf parking
functions}, and we call the permutation representation of $W$ on the set of
$W$-parking functions the (standard) $W$-{\sf parking space}. Parking spaces
have interesting connections to enumerative combinatorics, diagonal harmonics,
and rational Cherednik algebras. In this paper we define two new $W$-parking
spaces, called the {\sf noncrossing parking space} and the {\sf algebraic
parking space}, with the following features: 1) They are defined more generally
for real reflection groups. 2) They carry not just $W$-actions, but $W\times
C$-actions, where $C$ is the cyclic subgroup of $W$ generated by a Coxeter
element. 3) In the crystallographic case, both are isomorphic to the standard
$W$-parking space. Our Main Conjecture is that the two new parking spaces are
isomorphic to each other as permutation representations of $W\times C$. This
conjecture ties together several threads in the Catalan combinatorics of finite
reflection groups. We provide evidence for the conjecture, proofs of some
special cases, and suggest further directions for the theory.

Combinatorics