The betweenness centrality of a vertex of a graph is the fraction of shortest paths between all pairs of vertices passing through that vertex. In this paper, we study properties and constructions of graphs whose vertices have the same value of betweenness centrality (betweenness-uniform graphs); we show that this property holds for distance-regular graphs (which include strongly regular graphs) and various graphs obtained by graph cloning and local join operation. In addition, we show that, for sufficiently large $n$, there are superpolynomially many betweenness-uniform graphs on $n$ vertices, and explore the structure of betweenness-uniform graphs having a universal or sub-universal vertex.