This dissertation focuses on developing novel density-based topology optimization frameworks and optimization algorithms that can be used to obtain optimal structural designs while consistently incorporating the physics associated with higher-order continuum theories and inelastic constitutive models with plasticity/damage effects. Following are the key contributions of this thesis: Novel dual sequential approximation (DSA) algorithms for design variable updating are proposed for topology/structural optimization problems. These algorithms are demonstrated to have better performance on solving the benchmark problems.  Higher-order continuum theories – including the elasticity with microstructure, gradient elasticity and staggered gradient elasticity theories – are incorporated into the density-based topology optimization frameworks. These enhanced optimization frameworks can be used to design systems where material length-scale effects are prominent.  A topology optimization framework is proposed for design of energy absorbing elastoplastic systems under cyclic loads. In particular, the Bauschinger effect in materials under cyclic loads is considered using kinematic hardening rules. A unified framework for nonlinear path-dependent sensitivity analysis is developed that allows for consistent and accurate design sensitivity calculations with various inelastic models. The optimized designs demonstrate that Bauschinger effect has significant influence on the optimized topologies, and the designs are sensitive to the applied cyclic loading history.  Damage constrained topology optimization frameworks are proposed with the aim of optimally controlling the damage evolution in material during the elastoplastic energy dissipation process. Uncoupled, coupled and nonlocal elastoplastic-damage models are considered in the proposed topology optimization frameworks with the aim of balancing computational robustness and response accuracy. Effectiveness of the proposed frameworks is demonstrated through numerical examples, and the influence of adding damage constraints on the optimized designs is evaluated by parametric studies.  It is envisaged that the proposed topology optimization frameworks will lay the foundations for the next-generation of design methods wherein accurate physics of a system response is consistently considered and incorporated in the topology optimization design process.