The research is aimed at obtaining quantitative descriptions and critical thresholds of diffusive processes. One area of focus is mean first passage time problems in the presence of small absorbing traps (the narrow capture problem). A particular new direction of work is that in which the small traps are mobile, a scenario that arises in cellular process and predator-prey dynamics. Challenges include not only how to derive the correct PDE for such problems, but how to solve them using asymptotic and/or numerical methods. A critical question is under which condition(s) a mobile trap becomes more effective than a stationary one. The other major area of focus is the stability and dynamics of patterns in reaction-diffusion systems in one, two, and three spatial dimensions, both near and far from the linear regime. Here, asymptotic and numerical continuation techniques are used to construct and characterise steady-state patterns. Through a combination of analytic and numerical methods, critical thresholds are obtained for various types of instabilities. Delayed bifurcations of these instabilities, previously studied only in the context of ODE's near the linear regime, are also analyzed and their qualitative effects studied. The overarching theme is the techniques of analysis used (asymptotic methods, PDE techniques, numerical and continuation methods, Monte Carlo simulations), and that the quantitative results yielded by the analysis often lead to rich qualitative pictures of the underlying phenomena.