The development of computationally ecient algorithms for statistical inference of stochastic dierential equations has been a long-standing research subject ever since the advent of stochastic analysis. Recently, there has been an increasing interest in extending these methods to the statistical study of stochastic partial dierential equations. In this thesis, we are concerned with statistical inference for the linear stochastic convection-diusion equation (SCDE) driven by a stochastic forcing term that is spatially a Matern process. The approach is based on spectral methods for partial dierential equations and approximates a solution to the SCDE via a Fourier spectral decomposition. The resulting spectral processes are given by a family of uncoupled Ornstein-Uhlenbeck processes, for which computationally ecient statistical inference is possible based on the Kalman lter. We give veriable experimental results for all statistical problems - ltering, smoothing and parameter estimation - based on simulated data. We further derive a novel weak solution to the SCDE driven by spatial Matern noise for spatially periodic boundary conditions and show that the solution is indeed approximated by the Fourier spectral decomposition, thereby validating the statistical model. For the heat equation, we generalize the spectral ltering approach on compact Riemannian manifolds. Experimental results for this generalization are given on the two-dimensional unit sphere.