We review the use of phase field methods in solidification modeling, describing their fundamental connection to the physics of phase transformations. The inherent challenges associated with simulating phase field models across multiple length and time scales are discussed, as well as how these challenges have been addressed in recent years. Specifically, we discuss new asymptotic analysis methods that enable phase field equations to emulate the sharp interface limit even in the case of quite diffuse phase-field interfaces, an aspect that greatly reduces computation times. We then review recent dynamic adaptive mesh refinement algorithms that have enabled a dramatic increase in the scale of microstructures that can be simulated using phase-field models, at significantly reduced simulation times. Combined with new methods of asymptotic analysis, the adaptive mesh approach provides a truly multi-scale capability for simulating solidification microstructures from nanometers up to centimeters. Finally, we present recent results on 2D and 3D dendritic growth and dendritic spacing selection, which have been made using phase-field models solved with adaptive mesh refinement.