Summary: Reflected diffusions in convex polyhedral domains arise in a variety of applications, including interacting particle systems, queueing networks, biochemical reaction networks and mathematical finance. Under suitable conditions on the data, we establish pathwise differentiability of such a reflected diffusion with respect to its defining parameters – namely, its initial condition, drift and diffusion coefficients, and (oblique) directions of reflection along the boundary of the domain. We characterize the right-continuous regularization of a pathwise derivative of the reflected diffusion as the pathwise unique solution to a constrained linear stochastic differential equation with jumps whose drift and diffusion coefficients, domain and directions of reflection depend on the state of the reflected diffusion. Previous work in the multidimensional context has been largely restricted to the study of differentiability of stochastic flows for (normally) reflected Brownian motions. A key difficulty is to identify a suitable linearization of the dynamics of the local time process, especially in the presence of a non-smooth boundary. We take a new approach that uses properties of directional derivatives of the associated extended Skorokhod map, and their characterization in terms of the so-called derivative problem. The proof involves establishing certain path properties of the reflected diffusion at nonsmooth parts of the boundary of the polyhedral domain, which may be independent interest, and proving that pathwise derivatives of reflected diffusions can be characterized in terms of directional derivatives of the extended Skorokhod map. As a corollary, we obtain a probabilistic representation for derivatives of expectations of functionals of reflected diffusions, which is useful for sensitivity analysis of reflected diffusions.