I present a series of theoretical studies and techniques for modeling quantum dynamical systems. Recent technological developments in quantum simulators and quantum computers have given us a unique capability of controllably probing how a generic quantum system composed of many interacting constituents evolves in time. Inspired by such experimental capabilities, I analyze four different ways in which a quantum system can exhibit dynamics. The first two are examples of unitary dynamics which include cases where the initial quantum state is a non-stationary state of the Hamiltonian and where the Hamiltonian itself is time-dependent. The last two examples are of non-unitary dynamics where I focus on systems that evolve by dissipating energy/information to an environment and systems that stochastically evolve due to discrete non-commuting measurements. I study these different forms of quantum dynamics in experimentally motivated model situations. I use appropriate theoretical techniques and approximations to model each of these forms of quantum dynamics. In Chapter 2, I study the unitary dynamics resulting from the initial quantum state being in a non-stationary state (superposition of eigenstates) of the many-body Hamiltonian. Here I use a quantum Boltzmann equation to model a recent experiment that observed the evolution of a Bose-Einstein condensate from the highest to the lowest energy state in the excited band of an optical lattice. In Chapter 3, I consider unitary dynamics arising due to a time-dependent Hamiltonian. I study the dynamics of an interacting Bose gas in a rotating elliptical trap. By using a time-dependent variational wavefunction approach for my calculations, I explain how a recent experiment observed the rotating Bose gas entering a special quantum state, the lowest Landau level. In Chapter 4, I analyze non-unitary dynamics wherein a quantum system can dissipate its energy into the environment. I develop a protocol where the system and environment are engineered such that starting from a suitable initial state, the system goes to special target quantum states such as a Mott insulator state and a topologically ordered AKLT (Affleck-Kennedy-Lieb-Tasaki) state. I solve the Lindblad equation using exact diagonalization and DMRG methods to calculate these states' preparation timescales. Finally in Chapter 5, I study non-unitary dynamics that arise purely due to discrete quantum measurements of non-commuting operators. Motivated by the Bacon-Shor quantum error correcting code, I study the steady state ensemble obtained due to random measurements of nearest-neighbor XX and ZZ Pauli operators on qubits arranged on the vertices of a square lattice. I use the stabilizer formalism to efficiently represent the many-body quantum state and calculate the properties of the steady state phases as well as the criticality arising from this form of dynamics.