Let q be a prime power and suppose that e and n are integers satisfying 1 ⩽ e ⩽ n − 1. Then the Grassmann graph Γ(e, q, n) has as vertices the e-dimensional subspaces of a vector space of dimension n over the field Fq, where two vertices are adjacent iff they meet in a subspace of dimension e − 1. In this paper, a characterization of Γ(e, q, n) in terms of parameters is obtained provided that and ( if qϵ {2, 3}) and if q = 3). As a consequence we can show that these Grassmann graphs are uniquely determined as distance-regular graphs by their intersection arrays.