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Here, the focus is on codimension-two grazings that are local in the sense that the dynamics can be fully described by an appropriate Poincar\u00e9 map from a neighborhood of the grazing point (or points) of the critical cycle to itself. It is proposed that codimension-two grazing bifurcations can be divided into three distinct types: either the grazing point is degenerate, or the grazing cycle is itself degenerate (e.g. nonhyperbolic) or we have the simultaneous occurrence of two grazing events. A careful distinction is drawn between their occurrence in systems with discontinuous states, discontinuous vector fields, or that with discontinuity in some derivative of the vector field. Examples of each kind of bifurcation are presented, mostly derived from mechanical applications. For each example, where possible, principal bifurcation curves characteristic to the codimension-two scenario are presented and general features of the dynamics discussed. Many avenues for future research are opened. <\/jats:p>","DOI":"10.1142\/s0218127406015015","type":"journal-article","created":{"date-parts":[[2006,5,23]],"date-time":"2006-05-23T11:19:24Z","timestamp":1148383164000},"page":"601-629","source":"Crossref","is-referenced-by-count":98,"title":["TWO-PARAMETER DISCONTINUITY-INDUCED BIFURCATIONS OF LIMIT CYCLES: CLASSIFICATION AND OPEN PROBLEMS"],"prefix":"10.1142","volume":"16","author":[{"given":"P.","family":"KOWALCZYK","sequence":"first","affiliation":[{"name":"Department of Engineering Mathematics, University of Bristol, University Walk, Bristol BS8 1TR, UK"}]},{"given":"M.","family":"DI BERNARDO","sequence":"additional","affiliation":[{"name":"Department of Engineering Mathematics, University of Bristol, University Walk, Bristol BS8 1TR, UK"},{"name":"Dipartimento di Informatica e Sistematica, Universita degli Studi di Napoli, Naples, Italy"}]},{"given":"A. 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