In this paper, the first of a series dealing with low-dimensional lattices and their applications, we classify positive-definite integral quadratic forms of determinant d 25 in dimensions up to a limit which ranges from 18 (for d = 1) to 7 (for d = 25). [New version Nov. 5, 2000. A different version of this paper appeared in Proc. Royal. Soc. London, Series A, Vol. 418 (1988), pp. 17---41.] 1. Introduction This paper is the first in a series dealing with low-dimensional lattices and their applications. Here the application is to number theory: we classify all positive-definite integral quadratic forms of sufficiently small determinant and dimension. In later papers we intend to discuss maximal subgroups of GL(n , Z) (in Part II), perfect forms (in Part III), the mass formulae (in Part IV), and possibly other topics. One motivation for writing this series of papers is to present a systematic notation for these lattices. The need for this is illustrated by the fact that the symbols - ..