For a connected graph G of order n⩾3 and a set W⊆V(G), a tree T contained in G is a Steiner tree with respect to W if T is a tree of minimum order with W⊆V(T). The set S(W) consists of all vertices in G that lie on some Steiner tree with respect to W. The set W is a Steiner set for G if S(W)=V(G). The minimum cardinality among the Steiner sets of G is the Steiner number s(G). Connected graphs of order n with Steiner number n, n−1, or 2 are characterized. It is shown that every pair k,n of integers with 2⩽k⩽n is realizable as the Steiner number and order of some connected graph. For positive integers r,d, and k⩾2 with r⩽d⩽2r, there exists a connected graph of radius r, diameter d, and Steiner number k. Also, for integers n,d, and k with 2⩽d<n, 2⩽k<n, and n−d−k+1⩾0, there exists a graph G of order n, diameter d, and Steiner number k. For two vertices u and v of a connected graph G, the set I[u,v] consists of all vertices lying on some u– v geodesic in G. For U⊆V(G), the set I[U] is the union of all sets I[u,v] for u,v∈U. A set U is a geodetic set if I[U]=V(G). The cardinality of a minimum geodetic set is the geodetic number g(G). It is shown that g(G)⩽s(G) and that for every two integers a and b such that 3⩽a⩽b, there exists a graph G of radius r and diameter d such that d=r+1, g(G)=a, and s(G)=b.