The thesis consists of two main parts. In the first part, an alternative method of Monte Carlo simulation is presented where the expectation values are calculated by weighting configurations generated according to a multi-dimensional Gaussian distribution. The motivation for this approach is that any inefficiency resulting from not using importance sampling may be compensated by the rapid generation of Gaussian configurations. This is achieved by recursive filtering of arrays of Gaussian random numbers, hence the algorithm is called Linear Filtering. We found that the method offers substantial improvements in computer time for models where the action has a single minimum. To simulate more complicated models, the Gaussian must be replaced by distributions which respect the topology of the problem as well as being amenable to recursive filtering. We found distributions satisfying these requirements for two models with non-trivial topology. The other main problem considered is the computational study of the planar model. We present a method of simulating the Coulomb gas of vortices for the periodic Gaussian model which is the actual model used by Kosterlitz and Thouless in their analysis of the phase transitions in two-dimensional systems with continuous symmetry. The computational interest of the problem is the long range coupling between the vortices. In our method, the probabilities of a large number of vortex updates are calculated at each step and one update is selected using a procedure where the efficiency is independent of the peaking in the probabilities. The lack of dependence on peaking is in contrast to the usual Heat Bath algorithm and leads to small relaxation times. Our results are in good agreement with the predictions of the Kosterlitz and Thouless theory. We find the critical point at Tc = 1.39 and the specific heat peak at T= 1.65. The method is applicable to other models with long range coupling. Also considered in this thesis is the determination of the fractal dimension for quantum paths in one and two space dimensions with the aid of Monte Carlo simulations.