This thesis is in three parts. All parts are motivated by a desire to gain a better understanding of models of the phenomenon of two-dimensional diffusion limited aggregation, henceforth DLA. The first part proves some generalisations of results relating to Hastings-Levitov with α = 0, henceforth HL(0), another two-dimensional growth process. The second part is a study of numerical algorithms for simulating off-grid DLA. The third part describes and reports on some numerical experiments on multiple models of DLA. Part I provides a generalization of the concept of disturbance flows and of the coalescing Brownian flow, also known as the Brownian web, proving facts about the convergence of the former to the latter and about their time- reversals. This work was motivated as an attempt to generalize known results about the harmonic flow of HL(0) to the case of HL(2), which is supposed to be a model for DLA. Part II provides the first rigorous analysis of the asymptotic runtimes of four different previously published algorithms for simulating off-grid DLA. A variation on one of these algorithms, incorporating an improvement from another source and a trick new to this work, is implemented in code, with the runtimes comparing favourably to previous work. The runtime of this algorithm, like that of the algorithm it is based on, is Õ(n), which is optimal. Part III is a report on experiments testing whether or not off-grid DLA, HL(2) and noise-reduced DLA all have the same limiting shape in the many particle limit. It also contains a heuristic discussion of whether regularized HL can provide a good model for DLA. The results and heuristics indicate that regularizing HL with slit particles is not a promising way to simulate DLA. However, HL with circular particles, off-grid DLA and noise-reduced DLA are found to be in agreement.