The ideas in this thesis are placed broadly within the context of many-body quantum dynamics, an area of research that has gained significant interest in recent years due to developments in cold atom experiments that enable the realization of isolated many-body quantum systems. In this thesis, we first focus on the concept of connecting quantum mechanical systems to statistical mechanics, which often arises in the study of ‘thermalization’ in isolated many-body systems. An inescapable issue in the endeavor to connect the two is the definition of temperature. The first core definition of temperature we consider is inspired by the eigenstate thermalization hypothesis, which posits that the eigenstates of a generic thermalizing system have information regarding thermalization encoded within them. We consider temperatures based on comparing the structure of (full or reduced) eigenstate density matrices to thermal density matrices. The second temperature definition invokes the standard temperature-entropy relation from statistical mechanics relating temperature and microcanonical entropy. We explore various ways to define the microcanonical entropy in finite isolated quantum systems and numerically compute the corresponding temperature. Following this, we study the diametrical opposite of isolated quantum systems — open quantum systems. We study a quantum particle on a tight-binding lattice with a non-Hermitian (purely imaginary) local potential. Non-Hermitian Hamiltonians are effective models for describing open quantum systems. We analyze the scattering dynamics and spectrum, identifying an exceptional point where the entire spectrum pairs up into mutually coalescing eigenstate pairs. At large potential strengths, the absorption coefficient decreases, and the effect of the imaginary potential is similar to that of a real potential, which we quantify by utilizing the properties of a localized eigenstate. We demonstrate the existence of many exceptional points in a similar PT -symmetric system and non-interacting many-particle model. This investigation contributes to a many-body understanding of this non-Hermitian setup.