This thesis aims to showcase the versatility of statistical mechanics. It splits into two parts: firstly, applications of Doi-Peliti field theory to filament growth and branching processes, and secondly, wetting phenomena on structured surfaces. In Chapter 1, I derive the Doi-Peliti field theory; the derivation starts with stochastic processes that obey the Markov property and goes on to link them to the master equation. It is then recast as a partial differential equation of the probability generating function, the solution of which is found perturbatively with the Doi-Peliti path integral. This path integral formulation is used in Chapter 2 to model continuous-time branching processes. The critical behaviour of these processes is studied analytically and corroborated with simulations. The results were also published in [51] and show a natural link between the mathematics of stochastic processes and Doi-Peliti field theory. The field theory is then applied to the reaction-diffusion process of filament growth in Chapter 3. The filaments are polymers found in cells of living organisms, which assemble by incorporating their building blocks from the environment. However, if the environment does not contain enough building blocks, the filaments disassemble. The stochastic switching between disassembly and assembly leads to intriguing dynamics. Some of the results are part of a published article [118]. Statistical mechanics is also concerned with equilibrium phenomena. A class of these phenomena is studied in Chapter 4 in the context of wetting. Wetting occurs when a fluid forms a liquid film on a solid surface. If the surface is structured, i.e. has edges or corners, the occurrence of phase transitions of the liquid film depends on its specific structure. The structured surfaces studied in this thesis are the wedge and the apex; the results for them were also published in [133].