Summary: One method for communicating with multiple antennas is to encode the transmitted data differentially using unitary matrices at the transmitter, and to decode differentially without knowing the channel coefficients at the receiver. Since channel knowledge is not required at the receiver, differential schemes are ideal for use on wireless links where channel tracking is undesirable or infeasible, either because of rapid changes in the channel characteristics or because of limited system resources. Although this basic principle is well understood, it is not known how to generate good-performing constellations of unitary matrices, for any number of transmit and receive antennas and for any rate. This is especially true at high rates where the constellations must be rapidly encoded and decoded. We propose a class of Cayley codes that works with any number of antennas, and has efficient encoding and decoding at any rate. The codes are named for their use of the Cayley transform, which maps the highly nonlinear Stiefel manifold of unitary matrices to the linear space of skew-Hermitian matrices. This transformation leads to a simple linear constellation structure in the Cayley transform domain and to an information-theoretic design criterion based on emulating a Cauchy random matrix. Moreover, the resulting Cayley codes allow polynomial-time near-maximum-likelihood (ML) decoding based on either successive nulling/canceling or sphere decoding. Simulations show that the Cayley codes allow efficient and effective high-rate data transmission in multiantenna communication systems without knowing the channel.