Strongly coupled (or cross-diffusion) systems of parabolic and elliptic partial differential equations appear in many physical applications. This book presents a new approach to the solvability of general strongly coupled systems, a much more difficult problem in contrast to the scalar case, by unifying, elucidating and extending breakthrough results obtained by the author, and providing solutions to many open fundamental questions in the theory. Several examples in mathematical biology and ecology are also included. ContentsInterpolation Gagliardo-Nirenberg inequalitiesThe parabolic systemsThe elliptic systemsCross-diffusion systems of porous media typeNontrivial steady-state solutionsThe duality RBMO( u )- H 1( u )|Some algebraic inequalitiesPartial regularity