FM 96-12 Author: Giovanni Gallavotti Title: Methods in the theory of quasi periodic motions Abstract: Recent results on the theory of the Hamilton--Jacobi equation and the regularity of their solutions, in spite of the non regularity of the data, are discussed, with attention to the cancellation mechanisms that make regularity possible. Keywords: KAM, Quasiperiodic Schroedinger, Implicit functions Fisica, Universita' di Roma La Sapienza, P.le Moro 2, 00185, Roma, Italia. e-mail giovanni@ipparco.roma1.infn.it tel. 6-49914370, fax 6-4957697 Home page: http://www.math.rutgers.edu/~giovanni