The International Conference on Hyperbolic Problems: Theory, Numerics and Applications, 'HYP2008', was held at the University of Maryland from June 9-13, 2008. This was the twelfth meeting in the bi-annual international series of HYP conferences which originated in 1986 at Saint-Etienne, France, and over the last twenty years has become one of the highest quality and most successful conference series in Applied Mathematics. This book, the first in a two-part volume, contains nineteen papers based on plenary and invited talks presented at the conference. These original research and review papers written by leading experts as well as promising young scientists represent the state-of-the-art research frontiers in hyperbolic equations and related problems, ranging from theoretical analysis to algorithm development and applications in physical sciences and engineering. This volume will bring readers to the forefront of research in this most active and important area in applied mathematics.