Summary: This paper considers noncoherent communication over a frequency-nonselective channel in which the time-varying channel gain is unknown a priori, but is approximately constant over a coherence interval. Unless the coherence interval is large, coherent communication, which requires explicit channel estimation and tracking prior to detection, incurs training overhead which may be excessive, especially for multiple-antenna communication. In contrast, noncoherent detection may be viewed as a generalized likelihood ratio test (GLRT) which jointly estimates the channel and the data, and hence does not require separate training. The main results in this paper are as follows. (1) We develop a ``signal space'' criterion for signal and code design for noncoherent communication, in terms of the distances of signal points from the decision boundaries. (2) The noncoherent metric thus obtained is used to guide the design of signals for noncoherent communication that are based on amplitude/phase constellations. These are significantly more efficient than conventional differential phase-shift keying (PSK), especially at high signal-to-noise ratio (SNR). Also, known results on the high-SNR performance of multiple-symbol demodulation of differential PSK are easily inferred from the noncoherent metric. (3) The GLRT interpretation is used to obtain near-optimal low-complexity implementations of noncoherent block demodulation. In particular, this gives an implementation of multiple symbol demodulation of differential PSK, which is of linear complexity (in the block length) and whose degradation from the exact, exponential complexity, implementation can be made as small as desired