Graphene is not the only prominent example of two-dimensional (2D) materials. Due to their interesting combination of high mechanical strength and optical transparency, direct bandgap and atomic scale thickness transition-metal dichalcogenides (TMDCs) are an example of other materials that are now vying for the attention of the materials research community. In this article, the current state of quantum-theoretical calculations of the electronic and mechanical properties of semiconducting TMDC materials are presented. In particular, the intriguing interplay between external parameters (electric field, strain) and band structure, as well as the basic properties of heterostructures formed by vertical stacking of different 2D TMDCs are reviewed. Electrical measurements of MoS2, WS2, and WSe2 and their heterostructures, starting from simple field-effect transistors to more demanding logic circuits, high-frequency transistors, and memory devices, are also presented.