We propose a new approach for modelling non-linear multivariate interest rate processes based on time-varying copulas and reducible stochastic differential equations (SDEs). In the modelling of the marginal processes, we consider a class of non-linear SDEs that are reducible to Ornstein-Uhlenbeck (OU) process or Cox, Ingersoll, and Ross (1985) (CIR) process. The reducibility is achieved via a non-linear transformation function. The main advantage of this approach is that these SDEs can account for non-linear features, observed in short-term interest rate series, while at the same time leading to exact discretization and closed form likelihood functions. Although a rich set of specifications may be entertained, our exposition focuses on a couple of non-linear constant elasticity volatility (CEV) processes, denoted OU-CEV and CIR-CEV, respectively. These two processes encompass a number of existing models that have closed form likelihood functions. The transition density, the conditional distribution function, the steady-state density function are derived in closed form as well as the conditional and unconditional moments for both processes. In order to obtain more flexible functional form over time, we allow the transformation function to be time-varying. Results from our study of US and UK short term interest rates suggest that the new models outperform existing parametric models with closed form likelihood functions. We also find the time-varying effects in the transformation functions statistically significant. To examine the joint behaviour of interest rate series, we propose flexible non-linear multivariate models by joining univariate non-linear processes via appropriate copulas. We study the conditional dependence structure of the two rates using Patton (2006a) time-varying Symmetrized Joe-Clayton copula. We find evidence of asymmetric dependence between the two rates, and that the level of dependence is positively related to the level of the two rates.