In the last decade, Multiple Input Multiple Output (MIMO) radar has emerged as a leading candidate for stimulating major new advancement in radar theory. A fundamental challenge in MIMO radar is to identify a theoretical framework within which the radar system may be represented and analysed. In the relatively well-established field of Single Input Multiple Output (SIMO) array signal processing, this task has already been achieved using the array manifold (which is a geometric object that completely characterises the array system). A central objective of this thesis is therefore to bridge the gap between SIMO and MIMO by developing a manifold representation of the MIMO radar system. A new differential geometric framework, based on the complex Cartan matrix, is exploited in this thesis for characterising array manifold curves. New formulas are presented for recursively calculating the strictly orthonormal moving frame, U(s), and corresponding complex Cartan Matrix, C(s), for arbitrary array geometries. The circular approximation of the array manifold is derived under this new framework and compact closed-form expressions are provided for the popular uniform linear array geometry. Based on a number of approximations derived using the circular approximation of the array manifold, the performance capabilites of various popular detection and parameter estimation algorithms are investigated. The figure of merit "C" is then used to place these capabilities into the context of the theoretically ideal algorithm. The concept of a virtual SIMO array system is used as a basis for characterising the full MIMO radar configuration using a single equivalent response vector. By tracing out this response vector across the whole parameter space, a manifold is formed that fully characterises the MIMO radar system. In the important case of orthogonal transmit waveforms, the fundamental performance bounds of the MIMO radar system are studied. A space-time receiver architecture is proposed which exploits the virtual SIMO structure as part of a subspace-based joint Doppler, delay and direction of arrival (DOA) estimation framework. Due to the great computational burden of an exhaustive 3-parameter search, the joint Doppler-delay-DOA estimation is partitioned into an equivalent two-stage algorithm. The proposed approach is evaluated via computer simulation studies and shown to outperform existing methods.