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A subgraph of an edge-coloured graph is rainbow if all of its edges have different colours. For graphs G and H the anti-Ramsey number ar(G;H) is the maximum number of colours in an edge-colouring of
G with no rainbow copy of H. The notion was introduced by Erd}os, Simonovits and V. Sós and studied in case G=Kn. Afterwards exact values or bounds for anti-Ramsey numbers ar(Kn;H) were estab-
lished for various H among others by Alon, Jiang & West, Montellano-Ballesteros & Neumann-Lara, Schiermeyer. There are also results concernig bipartite graphs, cubes or product of cycles as G obtained by Axenovich, Li, Montellano-Ballesteros, Schiermeyer and others. In the talk the survey of these results will be given. Also results concerning anti-Ramsey numbers for cycles in complete split graphs will be
presented.
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