Quasiparticle band structures and optical properties of MoS2, MoSe2, MoTe2, WS2, and WSe2 monolayers are studied using the GW approximation in conjunction with the Bethe-Salpeter equation (BSE). The inclusion of two-particle excitations in the BSE approach reveals the presence of two strongly bound excitons (A and B) below the quasiparticle absorption onset arising from vertical transitions between a spin-orbit-split valence band and the conduction band at the K point of the Brillouin zone. The transition energies for monolayer MoS2, in particular, are shown to be in excellent agreement with available absorption and photoluminescence measurements. Excitation energies for the remaining monolayers are predicted to lie in the range of 1–2 eV. Systematic trends are identified for quasiparticle band gaps, transition energies, and exciton binding energies within as well as across the Mo and W families of dichalcogenides. Overall, the results suggest that quantum confinement of carriers within monolayers can be exploited in conjunction with chemical composition to tune the optoelectronic properties of layered transition-metal dichalcogenides at the nanoscale.