Monolayer group-VI transition metal dichalcogenides have recently emerged as semiconducting alternatives to graphene in which the true two-dimensionality is expected to illuminate new semiconducting physics. Here we investigate excitons and trions (their singly charged counterparts), which have thus far been challenging to generate and control in the ultimate two-dimensional limit. Utilizing high-quality monolayer molybdenum diselenide, we report the unambiguous observation and electrostatic tunability of charging effects in positively charged (X(+)), neutral (X(o)) and negatively charged (X(-)) excitons in field-effect transistors via photoluminescence. The trion charging energy is large (30 meV), enhanced by strong confinement and heavy effective masses, whereas the linewidth is narrow (5 meV) at temperatures <55 K. This is greater spectral contrast than in any known quasi-two-dimensional system. We also find the charging energies for X(+) and X(-) to be nearly identical implying the same effective mass for electrons and holes.