K.N. Majumdar has shown that for a 2−(v, k, λ) design there are three numbers α, τ, and Σ such that each intersection number of is not greater than Σ and not less than max{α, τ}. In this paper we investigate designs having one of these ‘extremal’ intersection numbers. Quasisymmetric designs with at least one extremal intersection number are characterized. Furthermore, we show that a smooth design having the intersection number Σ or α>0 is isomorphic to the system of points and hyperplanes of a finite projective space. Using this theorem, we can characterize all smooth strongly resolvable designs.