Block copolymers have emerged as a compelling candidate for nanoscale fabrication. In the context of lithographic fabrication, attention has focused on the ability to control the self-assembly and orientation of the nanodomains. However, assembling the block copolymers into defect-free structures can be difficult. We know defect-free structures have a low free-energy in equilibrium, but defects still occur in experiments. This shows that the defects are occurring due to the kinetics and dynamics of the block copolymers. In this thesis, I explore two methods to reduce defects via the Hole-Shrink experiments and shear, and the first steps of a dynamic model to reduce defects in chemoepitaxy was developed. In the second chapter, the graphoepitaxial assembly of cylinder-forming block copolymers assembled into holes is investigated through theoretically informed coarse grained Monte Carlo simulations (TICG MC). The aim is to identify conditions leading to assembly of cylinders that span the entire thickness of the holes, thereby enabling applications in lithography. Three hole geometries are considered, including cylinders, elliptical cylinders, and capsule-shaped holes. Four distinct morphologies of cylinder forming poly(styrene-b-methyl methacrylate) (PS-b-PMMA) block copolymers are observed in cylinders and elliptical holes, including cylinders, spheres, partial cylinders, and wall-bound cylinders. Additional morphologies are observed in capsule shaped holes. PMMA cylinders that extend through the entire hole are found with PMMA-wetting surfaces; a weak wetting condition is needed on the bottom of the hole and a strong wetting condition is necessary on the sides of the hole. Simulated are also used to explore the morphologies that arise when holes are overfilled, or when PMMA homopolymers are added in blends with copolymers. We find that overfilling can alter considerably the morphological behavior of copolymers in cylinders and, for blends; we find that when the homopolymer concentration is >10\%, the range of conditions for formation of PMMA cylinders that extend through the entire hole is increased. In general, results from simulations (TICG) are shown to be comparable to those of self-consistent (SCFT) calculations, except for conditions where fluctuations become important. In the third chapter, non-equilibrium simulations of lamellae forming block copolymers are investigated by means of Theoretically Informed Coarse Grain Brownian dynamics simulations and Dissipative Particle Dynamics. Three lamellar orientations are subjected to steady shear, which differ in the direction of the microstructure with respect to the shear plane. The stable orientations are identified as a function of shear rate. It is found that for Brownian dynamics simulations the transition from parallel to perpendicular does not occur; however, by including local conservation of momentum, the lamellae exhibit this transition. The velocity profiles, stresses, and angles of the blocks in the system were analyzed to yield insights into why parallel orientations are less stable at higher shear rates. In the fourth and fifth chapter, a model is developed for polymer entanglements in homopolymers and block copolymers. One possible approach relies on using slip springs. However, to inform these coarse grained models of block copolymers, the entanglements should be elucidated. Experiments do not provide the molecular data, so microscopic models are useful tools to study the molecular data. In this work, we study such issue using a Kremer-Grest model as it represents a "microscopic model". This model can be used to obtain the topological constraints of the polymer to inform coarse grained simulations. Defect annihilation of block copolymer nanodomains have been studied recently, and these studies have shown that there is a large kinetic barrier for defect annihilation. These studies has been performed with soft unentangled block copolymers, but in the experimental system, these block copolymers may be entangled. The entanglement effects of block copolymers are studied here using a slip spring model, which was previously developed for homopolymer melts. In this article, a comparison between the soft, slip spring model and a Lennard-Jones polymer is performed. The Lennard-Jones polymer is examined using the Z1 analysis, where it is shown that the topological constraints are increased at the interface of the unlike blocks for shorter chains, but for longer chains the constraints decrease at the interface. For the slip spring model, there is always a decrease of slip springs at the interface; when comparing the two models, we observe that large molecular soft chains mimic the Lennard-Jones polymers in both the topological constraints, but not the orientation of the chains near the interface.