This book provides the first comprehensive and complete overview on results and methods concerning normal frames and coordinates in differential geometry, with emphasis on vector and differentiable bundles. The book can be used as a reference manual, for reviewing the existing results and as an introduction to some new ideas and developments. Virtually all essential results and methods concerning normal frames and coordinates are presented, most of them with full proofs, in some cases using new approaches. All classical results are expanded and generalized in various directions. For example, normal frames and coordinates are defined and investigated for different kinds of derivations, in particular for (possibly linear) connections on manifolds, with or without torsion, in vector bundles and on differentiable bundles; they are explored also for (possibly parallel) transports along paths in vector bundles. Theorems of existence, uniqueness and, possibly, holonomicity of normal frames and coordinates are proved; mostly, the proofs are constructive and some of their parts can be used independently for other tasks. Numerous examples and exercises illustrate the material. Graduate students and researchers alike working in differential geometry or mathematical physics will benefit from this resource of ideas and results which are of particular interest for applications in the theory of gravitation, gauge theory, fibre bundle versions of quantum mechanics, and (Lagrangian) classical and quantum field theories.