. A theory is presented for simultaneousDiophantineapproximation by means of minimal sets of lattice points, defined in a certain sense. We show that successive minima can be used to find best simultaneous Diophantine approximations. We construct algorithms for producing these minima, and consequently best approximations, for lattices of the second degree and for lattices of the third degree. The latter algorithm is demonstrated with some numerical examples. 1. Introduction. This paper presents a theory regarding certain types of minimal sets of lattice points which are employed to produce algorithms for finding best simultaneous Diophantine approximations in a sense to be made precise. The definition of simultaneous Diophantine approximation we shall use is similar to that proposed by Lagarias [7], which is a generalisation of the more classical definitions, of which those given in [2] and [6] are examples. From this definition we produce a theory of (ae; h)-minimal sets of lattice p...