Rosenbloom and Tsfasman introduced a construction of lattices from algebraic curves over a finite field which have asymptotically good packing density in high dimensions. In recent joint work with Albrecht Boettcher, Lenny Fukshansky and Stephan Garcia, we define a class of lattices constructed from finite Abelian groups. I will talk about both constructions and their relationship. I will also talk about some geometric properties of these lattices. For example, if the lattice comes from an elliptic curve (unless the elliptic curve has exactly four points) or from a Hermitian function field, then it is generated by its set of minimal vectors. I will also talk about the number of minimal vectors and, if time permitting, about properties of the automorphism groups of these lattices.