The study of atomically-thin, truly two-dimensional (2D) materials has morphed into its own field since the experimental isolation of graphene and similar 2D materials in 2005. Graphene, as a single layer of carbon atoms with a unique band structure, and monolayer molybdenum disulfide (MoS₂), a three-atom-thick semiconductor, have been of particular interest both for the physics accessible in 2D crystals and the applications achievable with highly flexible materials. This dissertation presents a variety of experiments exploring the optoelectronic and mechanical properties of both monolayer MoS₂ and graphene. In particular, we present three studies: (1) the experimental realization of the valley Hall effect, an effect based on the Berry curvature of a material’s energy bands, in monolayer MoS₂; (2) methods for directly measuring the bending stiffness of graphene (and related 2D materials); and (3) an investigation of the wrapping of micro-spherical droplets by monolayer MoS₂. We conclude by discussing the future outlook of both "valleytronics" and microencapsulation by 2D materials.