Autonomous vehicles have the capability to revolutionize human mobility and vehicle safety. To prove safe, they must be capable of navigating their environment as well as or better than the best human drivers. The best human drivers can leverage the limits of a vehicle's capabilities to avoid collisions and stabilize the vehicle while sliding on pavement, ice, and snow. Automated vehicles should similarly be capable of navigating safety-critical scenarios when friction is limited, and one large advantage they hold over human drivers is the amount of data they can generate. With self-driving vehicles in the San Francisco Bay Area collecting almost two human lifetimes worth of data just during 2020, this abundance of data holds the key to improving vehicle safety. This dissertation examines how data generated by self-driving vehicles can be used to learn control policies and models to improve vehicle control near the limits of handling. As data collection and vehicle operation near the limits can be expensive, this work uses skilled humans as an inspiration for learning policies because of their incredible data efficiency. This ability is clearly demonstrated in racing where skilled human drivers act to improve their performance after each lap by shifting their braking point to maximize corner entry speed and minimize lap time. Starting from a benchmark feedforward and feedback control architecture already comparable to skilled human drivers, this work directly learns feedforward policies to improve vehicle performance over time. By using an approximate physics-based model of the vehicle, recorded lap data, and the gradient of lap time, this approach improves lap time by almost seven tenths of a second on a nineteen second lap over an initial optimization-based approach for racing. Additionally, this approach generalizes to low-friction driving. While model-based policy search shows improvement over a solely optimization-based approach, model-based policy search is ultimately limited by the vehicle model used. Physics-based models are useful for interpretability and understanding, but fail to make use of the abundance of data self-driving vehicles generate and often do not capture high-order or complex-to-model effects. Additionally, to operate at a vehicle's true limits, precise identification of the vehicle's road-tire friction coefficient is required which is a very difficult task. To overcome the drawbacks of physics-based models, this thesis next examines the ability of neural networks to use vehicle data to learn vehicle dynamics models. These models are capable of not only modeling higher-order and complex effects, but also vehicle motion on high- and low-friction surfaces. Furthermore, these models do so while retaining comparable control performance near the limits to a benchmark physics-based feedforward and feedback control architecture. Though this control approach shows promise in operating near the limits, feedforward and feedback control is ultimately limited in its ability to trade of small errors in the short term to prevent larger errors in the future. Additionally, actuator and road boundary constraints play an increasingly important role in safety as the vehicle nears the limits. To deal with these limitations, this work presents neural network model predictive control for automated driving near the limits of friction. Neural network model predictive control not only leverages the neural network model's ability to predict dynamics on high- and low-friction test tracks, but also retains comparable or better performance to MPC using a well-tuned physics model optimized to the corresponding high- or low-friction test track. While neural network MPC shows improved performance over physics-based MPC when operating near the limits, MPC leverages its dynamics model with complete certainty. These effects can lead to MPC overleveraging its dynamics model, which in the presence of model mismatch can lead to poor controller performance. Additionally, when using neural network models in MPC, the network predicts vehicle motion with complete certainty regardless of the presence or absence of training data in the corresponding modeled region. To mitigate this issue, this work presents an approach which leverages a neural network model to learn the uncertainty in the underlying dynamics model used in MPC. By learning the uncertainty in MPC's dynamics model, the vehicle can take actions to avoid highly uncertain regions of operation while still attempting to optimize the original MPC cost function. The insights from this work can be used to design automated vehicles capable of leveraging vehicle data to more effectively operate near the limits of handling.