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We consider the classical Inverse Function Theorem of Nash and Moser from the angle of some recent development by Ekeland and the authors. Geometrisation of tame estimates coupled with certain ideas coming from Variational Analysis when applied to a directionally differentiable function, produce very general surjectivity result and, if injectivity can be ensured, Inverse Function Theorem with the expected Lipschitz-like continuity of the inverse. We also present a brief application to differential equations., Comment: This is revised version. Implicit Mapping Theorem and an application to ODE were added

We provide a new proof along the lines of the recent book of A. Ioffe of a 1990's result of H. Frankowska showing that metric regularity of a multi-valued map can be characterized by regularity of its contingent variation - a notion extending contingent derivative.

We present simple and direct proof to an important case of Nash-Moser-Ekeland theorem., Comment: In this third version we have made numerous improvements inspired by the referee, like correcting the definition of directional derivative to the standard one and proving more general statement. The method of proof, however, is the same

We prove surjectivity result in Fr\'echet spaces of Nash-Moser type. That is, with uniform estimates over all semimorms. Our method works for functions which are only continuous and G\^ateaux differentiable like in the recent result of Ekeland. We present the results in multi-valued setting exploring the relevant notions of map regularity., Comment: The main correction in this version is clarifying that our definition of G\^ateaux differentiability is stronger than the standard one