Normally Hyperbolic Invariant Manifolds : The Noncompact Case Introduction Manifolds of bounded geometry Persistence of noncompact NHIMs Extension of results. This monograph treats normally hyperbolic invariant manifolds with a focus on noncompactness. These objects generalize hyperbolic fixed points and are ubiquitous in dynamical systems. First normally hyperbolic invariant manifolds and their relation to hyperbolic fixed points and center manifolds as well as overviews of history and methods of proofs are presented. Furthermore issues (such as uniformity and bounded geometry) arising due to noncompactness are discussed in great detail with examples. The main new result shown is a proof of persistence for noncompact normally hyperbolic invariant manifolds in Riemannian manifolds of bounded geometry. This extends well known results by Fenichel and Hirsch Pugh and Shub and is complementary to noncompactness results in Banach spaces by Bates Lu and Zeng. Along the way some new results in bounded geometry are obtained and a framework is developed to analyze ODEs in a differential geometric context. Finally the main result is extended to time and parameter dependent systems and overflowing invariant manifolds.