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We present a new nearby young moving group (NYMG) kinematic membership analysis code, LocAting Constituent mEmbers In Nearby Groups (LACEwING), a new Catalog of Suspected Nearby Young Stars, a new list of bona fide members of moving groups, and a kinematic traceback code. LACEwING is a convergence-style algorithm with carefully vetted membership statistics based on a large numerical simulation of the Solar Neighborhood. Given spatial and kinematic information on stars, LACEwING calculates membership probabilities in 13 NYMGs and three open clusters within 100 pc. In addition to describing the inputs, methods, and products of the code, we provide comparisons of LACEwING to other popular kinematic moving group membership identification codes. As a proof of concept, we use LACEwING to reconsider the membership of 930 stellar systems in the Solar Neighborhood (within 100 pc) that have reported measurable lithium equivalent widths. We quantify the evidence in support of a population of young stars not attached to any NYMGs, which is a possible sign of new as-yet-undiscovered groups or of a field population of young stars., Comment: 42 pages, 21 figures