  A lattice $\Lambda$ is said to be an extension of a sublattice $L$ of smaller
rank if $L$ is equal to the intersection of $\Lambda$ with the subspace spanned
by $L$. The goal of this paper is to initiate a systematic study of the
geometry of lattice extensions. We start by proving the existence of a
small-determinant extension of a given lattice, and then look at successive
minima and covering radius. To this end, we investigate extensions (within an
ambient lattice) preserving the successive minima of the given lattice, as well
as extensions preserving the covering radius. We also exhibit some interesting
arithmetic properties of deep holes of planar lattices.
