Advances in hardware design have made wheeled mobile robots (WMRs) exceptionally mobile. To fully exploit this mobility, WMR planning, control, and estimation systems require motion models that are fast and accurate. Much of the published theory on WMR modeling is limited to 2D or kinematics, but 3D dynamic (or force-driven) models are required when traversing challenging terrain, executing aggressive maneuvers, and manipulating heavy payloads. This thesis advances the state of the art in both the formulation and calibration of WMR models We present novel WMR model formulations that are high-fidelity, general, modular, and fast. We provide a general method to derive 3D velocity kinematics for any WMR joint configuration. Using this method, we obtain constraints on wheel ground contact point velocities for our differential algebraic equation (DAE)-based models. Our “stabilized DAE” kinematics formulation enables constrained, drift free motion prediction on rough terrain. We also enhance the kinematics to predict nonzero wheel slip in a principled way based on gravitational, inertial, and dissipative forces. Unlike ordinary differential equation (ODE)-based dynamic models which can be very stiff, our constrained dynamics formulation permits large integration steps without compromising stability. Some alternatives like Open Dynamics Engine also use constraints, but can only approximate Coulomb friction at contacts. In contrast, we can enforce realistic, nonlinear models of wheel-terrain interaction (e.g. empirical models for pneumatic tires, terramechanics-based models) using a novel force-balance optimization technique. Simulation tests show our kinematic and dynamic models to be more functional, stable, and efficient than common alternatives. Simulations run 1K-10K faster than real time on an ordinary PC, even while predicting articulated motion on rough terrain and enforcing realistic wheel-terrain interaction models. In addition, we present a novel Integrated Prediction Error Minimization (IPEM) method to calibrate model parameters that is general, convenient, online, and evaluative. Ordinarily system dynamics are calibrated by minimizing the error of instantaneous output predictions. IPEM instead forms predictions by integrating the system dynamics over an interval; benefits include reduced sensing requirements, better observability, and accuracy over a longer horizon. In addition to calibrating out systematic errors, we simultaneously calibrate a model of stochastic error propagation to quantify the uncertainty of motion predictions. Experimental results on multiple platforms and terrain types show that parameter estimates converge quickly during online calibration, and uncertainty is well characterized. Under normal conditions, our enhanced kinematic model can predict nonzero wheel slip as accurately as a full dynamic model for a fraction of the computation cost. Finally, odometry is greatly improved when using IPEM vs. manual calibration, and when using 3D vs. 2D kinematics. To facilitate their use, we have released open source MATLAB and C++ libraries implementing the model formulation and calibration methods in this thesis.