This volume presents a collection of problems and solutions in differential Geometry with applications. Both introductory and Advanced topics are introduced in an easy-to-digest manner, with the materials of the volume being self-contained. In particular, curves, surfaces, Riemannian and pseudo-riemannian manifolds, hodge duality operator, vector fields and lie series, differential forms, matrixvalued differential forms, maurer?cartan form, and the Lie derivative are covered. Readers will find useful applications to special and General relativity, yang?mills theory, hydrodynamics and field theory. Besides the solved problems, Each br>Chapter contains stimulating supplementary problems and software implementations are also included. The volume will not only benefit students in mathematics, applied Mathematics and theoretical Physics, but also researchers in the field of differential Geometry.