We extend the ideas of Barbour's paper from 1990 and adapt Stein's method for distributional approximation to infinite-dimensional distributions. Hence, we obtain theoretical results bounding the rate of functional convergence of certain classes of stochastic processes to diffusions. Those are applied to examples coming from queuing theory, random-graph theory, statistics and combinatorics. We firstly look at the motivation for this thesis and an overview of Stein's method. Then we present our original work, contained in four articles. The first one is a collaboration with Andrew Duncan and Sebastian Vollmer, published in the Electronic Communications in Probability and the other ones, for which I am the sole author, are currently under consideration for publication. The first paper corrects a mistake in Barbour's seminal work from 1990. The second paper considers the approximation of a time-changed Poisson process by a time-changed Brownian motion for time changes independent of the processes they are applied to. As an application, we study the M/M/1 queue and a time-changed Brownian Motion and bound a distance between the two. The third paper studies the asymptotic behaviour of scaled sums of random vectors having different dependence structures. As an application, a bound on the distance between scaled non-degenerate U-statistics and Brownian Motion is proved. Moreover, we prove a quantitative functional limit theorem for exceedances in the m-scans process. In the fourth paper, we adapt the exchangeable-pair approach to Stein's method to approximations by infinite-dimensional laws. It is used to provide the rate of convergence in a functional combinatorial central limit theorem, extending the result of Barbour and Janson from 2009. We further apply this approach to study the asymptotics of edge and two-star counts in a certain graph-valued process. The final part of the thesis presents the conclusions and suggestions for future work.</p
