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The maximum number of cusps on a plane algebraic curve of degree d is an open classical problem that dates back to the nineteenth century.A related open problem is the asymptotic value of the maximum number of cusps on plane curves of degree d, divided by d^2, when d tends to infinity.In this paper, we improve the best known lower bound forthe asymptotic value by constructing curves with the largest known number of cusps for infinitely many degrees. Some particular curves of relatively low degree with many cuspsare constructed too. The Appendix to this paper is devoted to the case of degree 11 and it is due to E. Shustin

Plane algebraic curves with many cusps, with an appendix by Eugenii Shustin