The cylindrical quasi-one-dimensional shape of colloidal semiconductor nanorods (NRs) gives them unique electronic structure and optical properties. In addition to the band gap tunability common to nanocrystals, NRs have polarized light absorption and emission and high molar absorptivities. NR-shaped heterostructures feature control of electron and hole locations as well as light emission energy and efficiency. We comprehensively review the electronic structure and optical properties of Cd-chalcogenide NRs and NR heterostructures (e.g., CdSe/CdS dot-in-rods, CdSe/ZnS rod-in-rods), which have been widely investigated over the last two decades due in part to promising optoelectronic applications. We start by describing methods for synthesizing these colloidal NRs. We then detail the electronic structure of single-component and heterostructure NRs and follow with a discussion of light absorption and emission in these materials. Next, we describe the excited state dynamics of these NRs, including carrier cooling, carrier and exciton migration, radiative and nonradiative recombination, multiexciton generation and dynamics, and processes that involve trapped carriers. Finally, we describe charge transfer from photoexcited NRs and connect the dynamics of these processes with light-driven chemistry. We end with an outlook that highlights some of the outstanding questions about the excited state properties of Cd-chalcogenide NRs.