The collocation method with weighted extended B-splines (WEB-splines) represents a recently published approach for the spline approximation of the solution of stationary partial differential equations. In contrast to standard finite element methods, WEB-collocation requires no mesh generation and numerical integration, which leads to considerably faster computation times and an easier implementation. In this talk, the basics of WEB-collocation for general boundary value problems with mixed boundary conditions are described and the advantages over finite element methods are illustrated for Poisson's equation as typical model problem. On this basis, current research results from the application to singular and time-dependent problems are presented. The utilization of uniform spline spaces permits a straightforward generalization of the basic concept to hierarchical bases and the development of intuitive refinement strategies. The benefits of these adaptive WEB-collocation algorithms are shown in case of the model problem with a singular solution. Furthermore, considering the problem of simulating a tsunami, the combination of the WEB-collocation concept and a time-step iteration is presented to demonstrate a novel approximation scheme for time-dependent equations.