The ability to anticipate the behavior of other vehicles on the road is a key part of how humans drive safely in complex environments. This thesis presents work enabling robotic systems to also anticipate the behavior of vehicles in the environment. The Hybrid Gaussian Mixture Model anticipation algorithm is presented, and enables the state of a dynamic system, such as a tracked vehicle, to be accurately predicted over useful time horizons, by using Gaussian Mixture Models to represent the state uncertainty, and adapting the Gaussian Mixture Models on the fly to any nonlinearities in the model of the dynamic system. Results show high accuracy predictions of a tracked vehicle state can be made in real time. The model used to anticipate the behavior of a vehicle in the environment must include both the vehicle dynamics and the driver behavior, so the Gaussian Process adaptive Gaussian Mixture Model (GP-aGMM) algorithm is presented, using Gaussian Processes to model human drivers and anticipate their behavior. Presented results show that the GP-aGMM can effectively anticipate the behavior of drivers even in complex situations. Finally, the lane-feature Gaussian Process Anticipation (LFGPG) algorithm is presented. The LFGPA algorithm is similar to the GP-aGMM, but abstracts the training data into a feature space that captures the relationship between the driver and the road, locally. This allows training data from one set of roads to be relevant to any roads. The power of the LFGPA algorithm to reduce the requirements for training data is demonstrated in presented results.