Block copolymers (BCP) are a class of materials that have attracted significant attention due to their ability to self-assemble into dense arrays of nanoscale features. These materials are being investigated for their use in applications such as nanolithography, but for commercial implementation require the ability to control or direct the self-assembly process. Chemoepitaxial directed self-assembly (DSA) is one avenue to achieving this control, where a BCP thin film self-assembles in the presence of precisely defined chemical boundary conditions. In such a process, the equilibrium structure of the BCP film and the kinetic pathways it evolves along to reach equilibrium are both a function of the thermodynamic landscape, which is in turn controlled by the chemical pattern. This thesis contributes to the significant body of work attempting to detail the relationship between chemical pattern parameters and the thermodynamics of assembly (both kinetic and equilibrium). We restrict our investigation to the assembly of lamellae-forming diblock copolymers on line/space chemical patterns that employ density multiplication, with a focus on developing technology for nanopatterning beyond the resolution limit of traditional lithography. In the first chapter we introduce the fundamental ideas of BCP DSA and develop the concepts of free energy balance that are crucial to framing the discussion in the following chapters. The second chapter explores using poly(methyl methacrylate) as a guide material and shows how the greater strength of guiding interaction for this system has the ability to guide complex, frustrated non-bulk morphologies. The third chapter develops a novel concept of using process conditions to generate so-called ‘three-tone’ chemical patterns with multiple guiding regions per patterned stripe. The fourth chapter looks at how guide stripe strength impacts and affects assembly kinetics, equilibrium structure, and process metrics such as line edge roughness (LER) and size of process window. It also introduces an analysis technique for evaluating assembly kinetics with an emphasis on defect annihilation. The fifth chapter seeks to identify more thoroughly the root causes of LER in BCP line/space DSA by investigating a number of factors. The sixth and final full chapter describes initial success in the effort to extend the concepts of BCP DSA on patterned planar substrates to flexible or three-dimensional substrates (for roll-to-roll applications) by using functional layer-by-layer deposited films. Our final conclusion touches on the ideas of nucleation of self-assembled BCP structures and how they relate to kinetic pathways and timescales of assembly.