In this paper we estimate a complete system of demand equations making full use of the restrictions implied by economic theory. Our theoretical model is based on the Klein-Rubin linear expenditure system which was first estimated by Stone. We place primary emphasis on maximum likelihood estimates obtained using annual time series observations of prices and per capita consumption for the U.S. economy in the period 1948-1965. The plan of the paper is as follows: Section 1 begins with a discussion of the problems involved in making systematic use of economic theory to estimate demand functions; this is followed by a brief description of the linear expenditure system and discussion of the specification of its dynamic and stochastic structure. In Section 2 we describe three methods of estimating the linear expenditure system, including the maximum likelihood procedure which we believe is most appropriate. We report our results in Section 3 and our conclusions in Section 4.