Many of the topological spaces involved in geometric topology, such as spaces of embedded manifolds, have complicated topologies that are difficult to specify and to manipulate. As already observed in the work of Galatius-Madsen-Tillmann-Weiss as well as Kupers, these spaces are often much easier to construct in the topos $Diff_{\leq 0}$ of set-valued sheaves on manifolds, and may moreover be endowed with naturally occurring smooth structures. Viewing $Diff_{\leq 0}$ as a subcategory of the infinity topos $Diff$ of homotopy-type valued sheaves - the eponymous differentiable sheaves - we will give conceptual proofs of how $Diff_{\leq 0}$ provides a model for the theory of homotopy types, and exhibit many good formal properties of $Diff_{\leq 0}$, such as the fact that all filtered colimits are homotopy colimits in $Diff_{\leq 0}$. By endowing $Diff$ with certain homotopical calculi, we are moreover able to obtain a generalisation of Berwick-Evans, Boavida de Brito, and Pavlov's result that for any (paracompact Hausdorff) manifold $A$, and homotopy-type valued Sheaf $X$ the mapping sheaf $Diff(A,X)$ computes the mapping space of the underlying homotopy types of $A$ and $X$.