In this thesis we have given an introduction to isogeometric finite element analysis on linear elasticity problems in 2D using non uniform rational B-splines (NURBS) as basis functions. We have studied the theory of B-splines and have derived the equations needed to perform linear elasticity stress analysis. An isogeometric finite element solver has been programmed in MATLAB. We have also analyzed the effect degenerated mappings have on the derivatives of the basis functions. We started by looking at a quadrilateral collapsing to a triangle, considering different parameterizations and their impact on the derivatives. We found that the derivatives were no longer in H^1 and that our basis was not a proper basis for finite element analysis. Our solution to this problem is to form a new set of basis functions by summing the basis functions at the singular points. Further we have applied this approach on a circular surface and an infinite plate with a circular hole.