  Define a "Liouville domain" to be a compact exact symplectic manifold with
contact-type boundary. We use embedded contact homology to assign to each
four-dimensional Liouville domain (or subset thereof) a sequence of real
numbers, which we call "ECH capacities". The ECH capacities of a Liouville
domain are defined in terms of the "ECH spectrum" of its boundary, which
measures the amount of symplectic action needed to represent certain classes in
embedded contact homology. Using cobordism maps on embedded contact homology
(defined in joint work with Taubes), we show that the ECH capacities are
monotone with respect to symplectic embeddings. We compute the ECH capacities
of ellipsoids, polydisks, certain subsets of the cotangent bundle of T2, and
disjoint unions of examples for which the ECH capacities are known. The
resulting symplectic embedding obstructions are sharp in some interesting
cases, for example for the problem of embedding an ellipsoid into a ball (as
shown by work of McDuff-Schlenk) or embedding a disjoint union of balls into a
ball. We also state and present evidence for a conjecture under which the
asymptotics of the ECH capacities of a Liouville domain recover its symplectic
volume.
