In this thesis the theory of depth functions is researched. Depth functions are functions that measure data depth and order multivariate observations. Two depth functions are discussed: the halfspace and simplicial depth function. The halfspace depth of a point is defined as the smallest probability for which a closed halfspace contains that point. The simplicial depth of a point is defined as the probability of that point being contained in a simplex for which its vertices are independent and identically distributed. Contours allow us to visualize these depth functions. This theory is applied to simulations with multivariate distributions and to weather statistics.