The aim of this thesis is to develop an algorithm for solving the best approximation problem in the L1 - norm with splines. Our study is based on the theory of L1-approximation with polynomials as well as theory that explain how this can be extended to splines. We will consider both the continuous and the discrete cases. Later we will also consider two practical algorithms for polynomial L1-approximation, based on Lagrange interpolation and linear programming. We will give an algorithm for computing so-called canonical points for splines, based on the Newtons method for finding zeros of a system of equations.