A theory of optimum receiver design for symbol-by-symbol detection of an uncoded digital data sequence received over the Gaussian channel with unknown carrier phase is presented. Linear suppressed-carrier modulation is assumed. The work here aims at laying a conceptual foundation for optimum symbol-by-symbol detection, and rectifies existing approaches to the problem. The optimum receiver structure is obtained explicitly for an arbitrary carrier phase model, but its computational requirements are too heavy in general for any practical implementation. In one important special case, namely, the case in which the carrier phase can be treated as a constant over some K+1 symbol intervals, the optimum receiver can be approximated by a readily implementable decision-feedback structure at high SNR. Simulated error performance results are presented for this latter receiver for PSK modulations, with various carrier phase models. Since a decision-feedback receiver can encounter a "runaway", a variation of this receiver is developed which uses feedforward of tentative decisions concerning future symbols. This modified receiver does not have any "runaway" problem, and has been shown to yield good error performance via simulations.>