This thesis explores several aspects of the correspondence between classes of linear ordinary differential equations (ODEs) in the complex plane and certain quantum integrable models (IMs), also known as the ODE/IM correspondence. First, we enlarge the set of ordinary differential equations that enter the correspondence. Differential equations satisfied by Wronskians between solutions of specific ODEs are obtained and are associated to nodes of particular Dynkin diagrams. In the second part of the thesis we generalise the correspondence to encompass massive IMs. Starting from an integrable nonlinear partial differential equation corresponding to the classical A2(l) affine Toda field theory (ATFT), we expand the set of integrable models that enter the correspondence. This establishes an ODE/IM correspondence for a massive IM. We then extend the results to the An-1 (1) ATFTs and the particular example of D3 (l) ATFT.