Semiconductor nanocrystals (NCs) are becoming increasingly entrenched in modern consumer technologies. Their versatility and capability to produce light emitting devices with high photoluminescence quantum yield, solar cells with excellent power conversion efficiencies, and extendibility to many other applications and industries including biotechnology have made NCs an exciting area of research and engineering. To realize the next generation of devices that use NCs effectively, it is important to understand how these nanoscale materials function at the quantum level using an array of techniques that can probe the electronic structure, morphology and size distributions, and ultrafast carrier dynamics that result in their remarkable macroscopic behavior. In this thesis I use two-dimensional electronic spectroscopy (2DES) to examine these properties and aid synthetic efforts to optimize these materials for various applications. In chapter 3, InP quantum dots are examined using many techniques includes 2DES to explain their broad emission spectra by understanding the electronic structure of emissive defects in the nanocrystalline lattice. Chapter 4 presents work in developing a novel synthesis of methylammonium lead-halide perovskites and using 2DES to probe the ultrafast dynamics of these NCs that are used in effective solar harvesting devices. In chapter 5, thin films of CdSe nanoplatelets are probed using 2DES with various pump laser powers, both at room temperature and 77K to gain insight into these materials that are used in low-threshold laser applications. Finally, chapter 6 concludes the thesis by proposing some future directions for our group as we continue to explore the quantum dynamics and electronic structure of interesting nanomaterials.