In this dissertation, we explore the effect of topology and geometry on equilibrium and out-of-equilibrium liquid crystals. In the first part, we study the geometry of nuclei of columnar lyotropic chromonic liquid crystals coexisting with an isotropic phase. Liquid crystal droplets coexisting in an isotropic environment show a rich morphology of droplet shapes, due to the delicate balance between the bulk and surface energy. Liquid crystal droplets are typically topologically spherical in shape to minimize their surface area; in this work, we study the nucleation of toroidal droplets that form when a columnar liquid crystal coexists with its own isotropic melt. We study a lyotropic chromonic liquid crystal (LCLC) that is formed by plank-like molecules of disodium cromoglycate (DSCG) with hydrophobic polyaromatic cores and hydrophilic peripheries. When dispersed in water, the DSCG molecules form cylindrical aggregates by stacking face-to-face. A high concentration of DSCG in water results in the appearance of the columnar (Col) phase, in which the aggregates arrange into a hexagonal lattice. We demonstrate that the toroidal shape of Col nuclei in the biphasic region depends strongly on the concentrations c of DSCG and C of a condensing agent polyethylene glycol (PEG). We explain the multitude of the observed shapes by the fine balance of bending elasticity and anisotropic interfacial tension. We also demonstrate that a droplet of a nematic liquid crystal confined between two glass plates changes its equilibrium shape from a simply-connected tactoid, which is topologically equivalent to a sphere, to a torus, which is not simply-connected. The topological transformation is triggered either by temperature or concentration and is explained by the interplay of nematic elastic constants, which facilitates splay and bend of molecular orientations in tactoids but prohibits splay in the toroids. In the second part, we explore out-of-equilibrium LCs formed by bacterial dispersions in patterned liquid crystals. We demonstrate that a nematic liquid crystal environment patterned as a spiral vortex controls the individual-to-collective transition in bacterial swirls and defines whether they expand or shrink. In dilute dispersions, the bacteria swim along open spiral trajectories following the pre-imposed molecular orientation. Above a certain concentration threshold, the bacteria condense into unidirectional circular swirls that resemble stable limit cycles. Their collective circular motion is controlled by the spiral angle of the vortex that defines the splay-to-bend ratio of the director, resulting in vortices with dominating splay to shrink toward the center and vortices with dominating bend to expand to the periphery. Spiraling vortices having an equal splay-to-bend ratio of the director produce stable swirls. The dynamic scenarios are explained by hydrodynamic interactions of bacteria mediated by the patterned passive nematic environment and by the coupling between the concentration and orientation.