Motivated by rapid improvements in the field of analog quantum simulation, I present a series of theoretical studies exploring exotic phases, phase transitions and transport properties in ultracold quantum matter. In Chapter 1, I describe the general goals of quantum simulation and survey recent developments in ultracold atom experiments, which motivate much of the work in this thesis. In Chapters 2 and 3, I investigate how strongly-correlated fermionic transport properties can be studied in ultracold atom quantum simulators. I identify two limits in which controlled calculations are possible. In Chapter 2 I use a quantum Boltzmann equation to study the weakly-interacting limit of the 2D Fermi-Hubbard model. In Chapter 3, I develop tensor network techniques to calculate transport coefficients in the strongly-interacting limit of the 1D mass-imbalanced Fermi-Hubbard model. In Chapter 4, I study the phases of strongly-interacting lattice bosons in one dimension using infinite matrix product states (iMPS). Harnessing spontaneous symmetry-breaking, I develop a novel technique for measuring the superfluid density using iMPS. In Chapter 5, I study the phases of frustrated and strongly-interacting lattice fermions with application to moire van der Waals heterostructures. Specifically, I study the zero-temperature phase transition from an insulating “electron crystal” to a metallic Fermi liquid, which is controlled by the ratio of the electron’s kinetic energy to its interaction energy. I find numerical evidence of two distinct pathways for the transition, involving either an intermediate superconducting phase or a direct continuous transition. In Chapter 6, I explore the efficiencies associated with spontaneous symmetry-breaking in iMPS simulations of gapless Luttinger liquids. I determine a general measure of efficiency that depends solely on the Luttinger parameter and present an extended discussion of conservation laws in iMPS.