- This paper consists of two parts, and in the first of these we develop the representation theory of solvable algebraic groups over a local field of characteristic zero in analogy with the work of Auslander and Eostant for solvable Lie groups. We show how all the representations arise and show that the Kirillov method of orbits applies to this situation. We find that the theory carries over completely and we discuss traces, CCR representations and we give a version of the Eostant independence of polarization theorem. In the second part we take up the problems of decomposing the space of square integrable functions on a solvable Lie group modulo a discrete cocompact subgroup. We show how to reduce this problem to the special case when the nilradical of the solvable group is Heisenberg. These two sections represent the initial part of a comprehensive program in this direction to be completed later. 1977, University of California, Berkeley. All Rights Reserved.