In magnetic resonance imaging (MRI), samples of the object's spectrum are measured in the spatial frequency domain (k-space). For a number of reasons there is a desire to reduce the time taken to gather measurements. The approach considered is to sample below the Nyquist density, using prior knowledge of the object's support in the spatial domain to enable full reconstruction. The two issues considered are where to position the samples (sampling strategies) and how to form an image (reconstruction techniques). Particular attention is given to a special case of irregular sampling, referred to as Cartesian sampling, in which the samples are located on a Cartesian grid but only constitute a subset of the full grid. A further special case is considered where the sampling scheme repeats periodically, referred to as periodic Cartesian sampling. These types of sampling schemes are applicable to 3-D Cartesian MRI, MRSI, and other modalities that measure a single point in 2-D k-space per echo. The case of general irregular sampling is also considered, which is applicable to spiral sampling, for example. A body of theory concerning Cartesian sampling is developed that has practical implications for how to approach the problem and provides intuition about its nature. It is demonstrated that periodic Cartesian sampling effectively decomposes the problem into a number of much smaller subproblems, which leads to the development of a reconstruction algorithm that exploits these computational advantages. An additional algorithm is developed to predict the regions that could be reconstructed from a particular sampling scheme and support; it can be used to evaluate candidate sampling schemes before measurements are obtained. A number of practical issues are also discussed using illustrative examples. Sample selection algorithms for both Cartesian and periodic Cartesian sampling are developed using heuristic metrics that are fast to compute. The result is a significant reduction in selection time at the expense of a slightly worse conditioned system. The reconstruction problem for a general irregular sampling scheme is also analysed and a reconstruction algorithm developed that trades off computation time for better image quality.