The author gives a survey on existing methods for calculating short vectors in a lattice and describes several applications. The underlying theory was developed in {\it D. E. Knuth} [The art of computer programming, Vol. 2 (1969; Zbl 0191.180)], the author [Math. Comput. 29, 827-833 (1975; Zbl 0306.10012)], {\it A. K. Lenstra}, {\it H. W. Lenstra} jun. and {\it L. Lovász} [Math. Ann. 261, 515-534 (1982; Zbl 0488.12001)], {\it U. Fincke} and the reviewer [Math. Comput. 44, 463-471 (1985; Zbl 0556.10022)]. Applications concern the investigation of pseudo-random numbers generated by the linear congruential method and diophantine approximation including a summary of the disproof of the Mertens conjecture by {\it A. M. Odlyzko} and {\it H. J. J. te Riele} [J. Reine Angew. Math. 357, 138-160 (1985; Zbl 0544.10047)].
Reviewer: M.Pohst