Fredholm properties of nonlocal differential operators via spectral flow Gregory FayeArnd Scheel 47G20Nonlocal operatorFredholm indexspectral flowNonlocal conservation lawEdge bifurcations. We establish Fredholm properties for a class of nonlocal differential operators. Using mild convergence and localization conditions on the nonlocal terms, we also show how to compute Fredholm indices via a generalized spectral flow, using crossing numbers of generalized spatial eigenvalues. We illustrate possible applications of the results in a nonlinear and a linear setting. We first prove the existence of small viscous shock waves in nonlocal conservation laws with small spatially localized source terms. We also show how our results can be used to study edge bifurcations in eigenvalue problems using Lyapunov-Schmidt reduction instead of a Gap Lemma. Indiana University Mathematics Journal 2014 text pdf 10.1512/iumj.2014.63.5383 10.1512/iumj.2014.63.5383 en Indiana Univ. Math. J. 63 (2014) 1311 - 1348 state-of-the-art mathematics http://iumj.org/access/