The familiar macroscopic properties of matter emerge out of complex dynamics occurring at the molecular scale. Establishing this connection between microscopic and macroscopic behaviors is the principal task of statistical mechanics. But collective phenomena can emerge out of individual motion and interactions in a much broader class of systems than those with which statistical mechanics has traditionally been concerned, including systems composed of particles that move under their own power by consuming energy from their environment. Known as active matter, such systems can exhibit novel phase and transport behavior with both surprising similarities and striking differences compared to ordinary matter. In this dissertation, I present contributions to the development of a revised statistical mechanical framework describing the emergence of linear transport phenomena in active matter. In particular, odd transport phenomena, in which fluxes arise in the direction perpendicular to thermodynamic driving forces, are shown to be a consequence of chirality in the microscopic fluctuations, characterized by the breaking of time-reversal and parity symmetries. The main results presented in this dissertation consist of the derivations of Green-Kubo relations connecting microscopic symmetries to macroscopic odd transport, together with numerical validation of these relations through molecular dynamics simulations of active model systems. Through this lens, odd diffusion and odd viscosity are introduced and developed. I conclude by presenting a general framework for deriving Green-Kubo relations for odd transport coefficients in active matter. Taken as a whole, these results provide a fruitful extension of existing statistical mechanics concepts, facilitating an understanding of the microscopic origins of odd transport phenomena and indicating the physical contexts in which new types of odd transport can be expected.