The Shi and Catalan arrangements are deformations of the braid arrangement with remarkable numerological and structural properties. Their regions can be labeled by parking functions and trees, and natural statistics on their dominant regions determine the Whitney numbers for the lattice of noncrossing partitions. These results generalize to all Weyl groups W.
I will present recent work generalizing these constructions to restrictions A^X of reflection arrangements A on arbitrary flats X. The simple arrangements A^X are not crystallographic but we are able to construct free deformations for them with similar numerological and structural results as the Catalan and Shi cases. We will further connect them to the representation theory of parking spaces and rational Cherednik algebras, to the combinatorics of W-noncrossing partitions, and to ramification formulas associated to the braid monodromy of the Weyl groups W.