Weather derivatives are a new type of financial contract that derive their value from weather measurements over the period of the contract. In this thesis we focus on temperature-based Cooling Degree Day (CDD) and Heating Degree Day (HDD) contracts, developing continuous time stochastic mean reversion models for temperature based on the Ornstein-Uhlenbeck process. In the first half of the thesis, we consider models with Brownian motion as the driving noise, firstly in the scalar case, both for constant volatility and for seasonal volatility, and then generalize to the vector case in order to simultaneously model several correlated weather variables. In each case, we fit the models to temperature data from Brisbane and Melbourne. We go on to develop methods of estimating the price of some CDD and HDD contracts, comparing the estimates with historical simulations. In the second half of the thesis, we extend the model to include Poisson jumps. We develop several methods for approximating the Fourier transforms that give the transition probability densities, comparing the efficiency and accuracy of each, and fit the model parameters to Brisbane and Melbourne temperature data. We conclude by pricing some CDD and HDD contracts under the jump model. The techniques developed in this thesis are valuable in that they are applicable to a much more general range of problems, including interest rates and commodity prices.