Neutron stars are the superdense remnants of massive stars that have perished in supernovae. These astrophysical objects are invaluable for the study of matter at low temperatures and supranuclear densities, as these conditions cannot currently be recreated by terrestrial experiments. The possible stable existence of exotic matter in the neutron star core is an active area of research and the focus of this investigation. The exotic matter considered in this work includes hyperons, delta isobars, and deconfined quarks. To begin we introduce the relativistic mean-field approximation (RMF) that is used in this work for modeling hadronic neutron star matter. We determine which popular RMF parameterizations produce results most compatible with the current constraints from nuclear physics and neutron star observations. Next we include hyperons and delta isobars in the hadronic composition, compute the equation of state, and determine if the resulting neutron star properties are consistent with astrophysical constraints. In addition, we systematically explore the parameter space of the meson-hyperon and meson-delta coupling constants. The results of this exploration are used to constrain the coupling constants and analyze their effect on the population of exotic baryons in neutron star matter. Finally, we construct quark-hadron hybrid equations of state using the nonlocal three-flavor Nambu-Jona-Lasinio model for deconfined quark matter and both the Maxwell and Gibbs conditions for phase equilibrium. We find current constraints do not rule out exotic neutron star matter, and that hyperons, delta isobars, and deconfined quarks could all coexist in the neutron star core.