  Since the Hadamard Theorem, several metric and topological conditions have
emerged in the literature to date, yielding global inversion and implicit
theorems for functions in different settings. Relevant examples are the
mappings between infinite-dimensional Banach-Finsler manifolds, which are the
focus of this work. Emphasis is given to the nonlinear Fredholm operators of
nonnegative index between Banach spaces. The results are based on good local
behavior of $f$ at every $x$, namely: $f$ is a local homeomorphism or $f$ is
locally equivalent to a projection. The general structure includes a condition
that ensures a global property for the fibres of $f$, ideally, expecting to
conclude that $f$ is a global diffeomorphism or equivalent to a global
projection. A review of such these results and some relationships between
different criteria are shown. Also, in this context, a global version of Graves
Theorem is obtained.
