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We investigate the maximal open domain $\mathscr{E}(M)$ on which theorthogonal projection map $p$ onto a subset $M\subseteq \mathbb{R}^d$ can bedefined and study essential properties of $p$. We prove that if $M$ is a $C^1$submanifold of $\mathbb{R}^d$ satisfying a Lipschitz condition on the tangentspaces, then $\mathscr{E}(M)$ can be described by a lower semi-continuousfrontier function. We show that this frontier function is continuous if $M$ is$C^2$ or if the topological skeleton of $M^c$ is closed and we provide anexample showing that the frontier function need not be continuous in general.  We demonstrate that, for a $C^k$-submanifold $M$ with $k\ge 2$, theprojection map is $C^{k-1}$ on $\mathscr{E}(M)$, and we obtain adifferentiation formula for the projection map which is used to discussboundedness of its higher order derivatives on tubular neighborhoods.  A sufficient condition for the inclusion $M\subseteq\mathscr{E}(M)$ is that$M$ is a $C^1$ submanifold whose tangent spaces satisfy a local Lipschitzcondition. We prove in a new way that this condition is also necessary. Moreprecisely, if $M$ is a topological submanifold with $M\subseteq\mathscr{E}(M)$,then $M$ must be $C^1$ and its tangent spaces satisfy the same local Lipschitzcondition.  A final section is devoted to highlighting some relations between$\mathscr{E}(M)$ and the topological skeleton of $M^c$.

Existence, Uniqueness and Regularity of the Projection onto Differentiable Manifolds