Both colloidal particles and polymers often form ordered liquid phases that posses unique optical, rheological, mechanical, or biological properties, making them attractive components in the preparation of novel fibers, coatings, nanocomposites, and bio-active materials. Consistent with the recent advances in synthetic techniques that are realizing novel mesogenic building blocks with unprecedented degree of control, our goal is to use Monte Carlo simulations to map out the still uncharted phase behavior of systems containing such building blocks. In this talk I will first present some of the recent advances in my group regarding methodologies for simulating free energies and density of states. I will then present results for several systems exhibiting order-disorder transitions including a suspension of cuboids (a model of colloidal particles), a system of diblock copolymers, a polymer network, and an antibody binding site. In the system of cuboids, our results show the presence of novel entropy-driven liquid-crystalline phases arising from varying the aspect ratio of the cuboid (going from square tiles to long rods). In the diblock copolymer we elucidate the role that entropic forces play in limiting the stability of co- and bi-continuous phases and the effect of added nanoparticles. Our results for a model entanglement-free, semiflexible-chain network show an unusual step-wise elastic behavior, reminiscent of that observed in super-tough natural materials like titin. Finally, our results for a llama antibody fragment illustrate the importance of inter-loop interactions (besides intra-loop interactions) in determining the structure of atypical 3-loop antibody binding sites.