  In semiconductor physics, many essential optoelectronic material parameters
can be experimentally revealed via optical spectroscopy in sufficiently large
magnetic fields. For monolayer transition-metal dichalcogenide semiconductors,
this field scale is substantial --tens of teslas or more-- due to heavy carrier
masses and huge exciton binding energies. Here we report absorption
spectroscopy of monolayer MoS$_2$, MoSe$_2$, MoTe$_2$, and WS$_2$ in very high
magnetic fields to 91~T. We follow the diamagnetic shifts and valley Zeeman
splittings of not only the exciton's $1s$ ground state but also its excited
$2s$, $3s$, ..., $ns$ Rydberg states. This provides a direct experimental
measure of the effective (reduced) exciton masses and dielectric properties.
Exciton binding energies, exciton radii, and free-particle bandgaps are also
determined. The measured exciton masses are heavier than theoretically
predicted, especially for Mo-based monolayers. These results provide essential
and quantitative parameters for the rational design of opto-electronic van der
Waals heterostructures incorporating 2D semiconductors.

Mesoscale and Nanoscale Physics;Materials Science