This monograph presents most up-to-date developments and results in almost periodicity and almost automorphy in abstract spaces. It discusses various traditional and several new methods, including the methods of invariant subspaces and uniform spectrum, to obtain almost periodic and almost automorphic solutions to some linear and non-linear evolution equations and dynamical systems. The book also discusses fuzzy-number type spaces and their applications to differential equations. Enriched by excellent open problems that offer a starting point for further research, the text is an invaluable reference for graduate students and researchers working in Analysis.