Since 1994, after the first meeting on "Quaternionic Structures in Mathematics and Physics", interest in quaternionic geometry and its applications has continued to increase. Progress has been made in constructing new classes of manifolds with quaternionic structures (quaternionic Kaehler, hyper Kaehler, hyper-complex, etc), studying the differential geometry of special classes of such manifolds and their submanifolds, understanding relations between the quaternionic structure and other differential-geometric structures, and also in physical applications of quaternionic geometry. Some generalizations of classical quaternion-like structures (like HKT structures and hyper-Kaehler manifolds with singularities) appeared naturally and were studied. Some of those results are published in this book. This book should be of interest to researchers and graduate students in geometry, topology, mathematical physics and theoretical physics.