The minimal surface equation is an elliptic equation but it is nonlinear and is not uniformly elliptic. It is the Euler-Lagrange equation for variational problems which involve minimising the area of the graphs of functions. For the most part we will solve the variational problem with Dirichlet boundary values, that is, when the values of the function are prescribed on the boundary of some given set. We will present some existence results using the Direct Method from the Calculus of Variations and also some interior gradient estimates. All of the techniques can be generalised to include more difficult equations but the essence of the ideas is much clearer when dealing with this particular equation especially as it has such strong geometrical meaning. The material presented closely follows Chapters 12 and 13 from the book "Minimal Surfaces and Functions of Bounded Variation" by E. Giusti.