Signal design theory is concerned with the problem of determining transmitter signal waveforms such that the probability of correct reception (or some other appropriate measure of communication efficiency) is maximized. The selection of signals must be performed under specified constraints of signal power and bandwidth and channel noise disturbance. In this paper three cases are treated for the coherent Gaussian channel: \begin{enumerate} \item white noise, equal signal energies, unequal message probabilities, \item white noise, average signal power bounded, equal message probabilities, and \item colored noise, average signal power bounded, equal message probabilities. \end{enumerate} In each problem, necessary general conditions for signal optimality are derived, and specific solutions obtained for small and large signal-to-noise ratios (SNR's). Complete solutions are indicated for the case of three messages. It is shown that the regular simplex codes are always global solutions at large SNR. Other results are also presented.