We investigate phase stability in all binary alloys comprised of elements from groups 4 (Ti, Zr, Hf), 5 (V, Nb, Ta) and 6 (Cr, Mo, W) of the periodic table. First-principles calculations of the energy landscapes along crystallographic pathways that connect bcc to hcp and bcc to ω show that group 4 elements are very distinct from group 5 and 6 elements. While group 5 and 6 elements are stable in bcc, group 4 elements favor hcp and ω and are predicted to be dynamically unstable in bcc. A comprehensive first-principles investigation of the 36 refractory binary systems using statistical mechanics techniques reveals six distinct classes of alloys, each with a unique phase diagram topology. The predictions of this study are in excellent agreement with previous experimental work. One exception is a class of refractory alloys with high temperature miscibility gaps that are not predicted with the methods used in this work. Our calculations predict the stability of a low-temperature Laves phase in the Nb-V binary that has yet to be observed experimentally. The relationships between alloy chemistry and high-temperature phase stability revealed in this study provide a basis for the systematic design of multicomponent disordered refractory alloys.