This thesis is concerned with the development and extension of various new approaches to solving the Coupled Cluster (CC) equations through Monte Carlo (MC) integration. Firstly, exact importance sampling of the CC wavefunction to arbitrary order is defined and applied as a modification to the extant unlinked Coupled Cluster Monte Carlo (CCMC) method. This is shown to be vital for the stability of the method while reducing both computational and memory costs on test calculations. Stochastic approaches utilising the diagrammatic formalism of deterministic CC theory in its unfactorised and totally factorised forms are then developed. These are found to show reduced computational costs compared to prior approaches. It is demonstrated that sampling only the connected components of the similarity-transformed Hamiltonian obtains a solution with memory cost proportional to system size for perfectly local systems and a fixed granularity of representation. For a fixed errorbar per replica calculation costs scale as quartic and linear in the number of replicas for the unfactorised and factorised approaches, respectively. This behaviour is independent of truncation level, and shown to translate into approximately local physical systems. The presence of various systematic biases resulting from wavefunction nonlinearity within these approaches is then investigated. All metrics introduced are found to be negligible compared to the accuracy required for chemical applications in an example system, and possible reasons for this suggested. By considering developments of Full Configuration Interaction Quantum Monte Carlo (FCIQMC) extensions of the totally factorised approach are formulated based upon the semistochastic and initiator approximations, and are shown to provide reduced computational and memory costs. Finally, these new approaches are combined with probabilistic considerations to allow exact solutions to the CC equations to be obtained while only considering approximate update steps.