In this report we discuss linear Model Predictive Control (MPC) from a computational and algorithmic perspective. We describe two commonly encountered MPC formulations; the first includes the system model as an explicit equality constraint while the second includes the system model implicitly by projecting onto the sub-space described by this linear relation. Both formulations are expressed as Quadratic Programming (QP) problems. We discuss a popular primal-dual predictor-corrector interior-point algorithm used in solving QP problems. We go on to discuss how the interior-point strategy can exploit the structure of MPC when solving a linear sub-problem found within such algorithms. This leads to the well known Riccati recursion approach which has associated linear complexity in the prediction horizon. A simulation example is used to illustrate that the explicit formulation of MPC may indeed exhibit this property even without using a Riccati recursion.