In this thesis, we study wireless communication systems equipped with multiple antennas at the transmitter and the receiver. In particular, we investigate the design and performance analysis of transmission schemes when partial channel state information is available at the transmitter via a feedback channel. Assuming that the feedback information is used for beamforming, we study the design of the transmission schemes and the feedback channel with the finite rate feedback constraint. The important contribution of this thesis is a geometrical framework for studying beamforming schemes for multiple antenna systems in the presence of finite rate channel feedback. We present intuitive geometrical interpretations to the beam forming problem and demonstrate the utility of the geometrical framework through the derivation of a fundamental lower bound on the error performance of beam-forming schemes. Using the tight lower bound, we present a distortion measure for quantization based on the resulting outage probability which bears interesting similarities with the mean squared distortion metric commonly used in source coding. In particular, we show that finite rate feedback systems approach the perfect channel information case as t-12-B t-1 for a single receive antenna where B is the number of feedback bits and t is the number of transmit antennas. Our geometrical framework also leads to a design criterion for good beamformers. Our design criterion for beamformer construction minimizes the maximum inner product between any two beamforming vectors in the beamformer codebook. We show the equivalence of beamformer construction with two other interesting problems in literature, viz., unitary space time code design for non-coherent communications and the subspace packing problem. We show that good beamformers are good packings of 2-dimensional subspaces in a 2t-dimensional real Grassmannian manifold with chordal distance as the distance metric. Finally, we extend our analysis and construction methods to multiple receive antenna systems. We introduce the notion of partial rank beamforming and derive a rate region where rank-m beamforming can be expected to perform better than the full rank space time codes. Rate threshold Rth,m, which is the maximum of the rates in the optimal rate region of rank-m beamforming, is shown to vary as kmk-mlog 2&parl0;tm&parr0; , where k = min(r, t), r being the number of receive antennas. (Abstract shortened by UMI.)