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In this article we apply the classical method of focal loci of familiesto give a lower bound for the genus of curves lying on general surfaces. Firstwe translate and reprove Xu's result that any curve C on a general surfacein P3 of degree d  5 has geometric genus g > 1 + degC(d-5)=2. Then weprove a similar lower bound for the curves lying on a general surface in a given component of the Noether-Lefschetz locus in P3 and on a general projectively Cohen-Macaulay surface in P4

Focal loci of families and the genus of curves on surfaces