The Casimir force is a force between neutral bodies derived entirely from quantum fluctuations of the electromagnetic field. In this talk I will start by discussing some of the events leading up to Hendrik Casimir's prediction of the existence of this force in 1948. I will then discuss the mathematical modelling of the force and describe mode summation, which is the most common method for computing the Casimir force. I will point out it's well known limitations and then discuss the main two numerical approaches to the Casimir problem. The first is based on a functional integral formulation of Quantum Field Theory and the second on partial differential equations for the field correlator. In my discussion of both approaches I will limit myself for technical reasons to the case of a neutral scalar field.