  In a series of previous papers, we studied sortable elements in finite
Coxeter groups, and the related Cambrian fans. We applied sortable elements and
Cambrian fans to the study of cluster algebras of finite type and the
noncrossing partitions associated to Artin groups of finite type. In this
paper, as the first step towards expanding these applications beyond finite
type, we study sortable elements in a general Coxeter group W. We supply
uniform arguments which transform all previous finite-type proofs into uniform
proofs (rather than type by type proofs), generalize many of the finite-type
results and prove new and more refined results. The key tools in our proofs
include a skew-symmetric form related to (a generalization of) the Euler form
of quiver theory and the projection \pidown^c mapping each element of W to the
unique maximal c-sortable element below it in the weak order. The fibers of
\pidown^c essentially define the c-Cambrian fan. The most fundamental results
are, first, a precise statement of how sortable elements transform under (BGP)
reflection functors and second, a precise description of the fibers of
\pidown^c. These fundamental results and others lead to further results on the
lattice theory and geometry of Cambrian (semi)lattices and Cambrian fans.
