Convection-reaction-diffusion equations can describe a diverse range of physical phenomena. The development of efficient, reliable, and accurate numerical methods for the solution of such equations is ongoing, especially for certain types of problems (e.g. ones in which convection dominates). In this thesis, a new method, called the Lumped-component Circuit Method (LCM), developed previously for one-dimensional steady-state reaction-diffusion, is tested and extended for modelling both steady-state and transient reaction-diffusion and convection-reaction-diffusion in one and two dimensions. It is developed for solving equations with piecewise-constant coefficients, but its application is not restricted to such problems. Like the Transmission Line Method (TLM), it is an indirect method in which the problem to be solved is first represented by an analogous transmission line (TL). Unlike with TLM, however, the TL is then modelled using a lumped-component circuit, and the voltages at nodes within that circuit are calculated. For transient modelling, a time-stepping scheme is required. Traditional schemes can be used when calculating the node voltages over time, but TLM (a simple, explicit, and unconditionally stable time-stepping technique) can also be used for this purpose. The LCM method is compared with FVM (Finite Volume Method) schemes. It is validated, where possible, using analytical solutions and existing solutions to real physical problems. When solving equations with piecewise-constant coefficients, with nodes that are not positioned to correspond with the discontinuities, the FVM solutions do not converge consistently as the node spacing is decreased. That is not the case with LCM. In general, the LCM scheme is more accurate than the FVM schemes tested, and, while the computational cost of LCM is higher, results suggest that it is generally more accurate, especially when one or more of the coefficients are piecewise constant.