This work is an expository overview of certain key elements in the area of compressive sensing. As a sub-discipline of signal processing, compressive sensing is concerned with both sampling and reconstruction techniques. In this expository, sampling will center on random matrices and expander graphs, while reconstruction will use multiple numerical optimization techniques. Although theoretical performance bounds for these techniques can be found scattered throughout the published literature, there are few practical rules for concrete problems. This thesis helps fill this gap by experimenting on the asymptotic bounds of the number of measurements needed to guarantee perfect reconstruction. These numerical experiments help to identify specific sensing regimes in which performance begin to break down
