Abstract: Jump systems were discovered in 1995 by Bouchet and Cunningham. A jump system is a subset of the integer lattice satisfying a single simple axiom. They can be used to model objects such as matroids and graph degree sequences. Two natural problems are the membership problem (is a given lattice point a member of an abstractly specified jump system) and the intersection problem (do two jump systems have a nonempty intersection). It turns out that these problems are equivalent! This talk will present a survey of progress toward these and other problems.