I will review recent advances in our understanding of transport in mesoscopic strongly correlated electron systems in one dimension. Prominent examples of these systems are semiconductor quantum wires, carbon nanotubes, and coupled quantum Hall edges running in opposite directions. I will focus on the interplay between the effects of disorder and interactions; specifically, on the effects of Anderson localization in a disordered Luttinger liquid. In the presence of disorder, the conductivity at finite temperature is nonzero only because of inelastic electron-electron scattering. I will demonstrate that the notions of weak localization and dephasing due to Coulomb interactions are applicable to the strongly correlated 1D system. The dephasing rate depends strongly on the spin polarization and for spinless electrons is totally different from the Aharonov-Bohm dephasing rate. With lowering T, the system exhibits localization transition in many-body Fock space ("Anderson-Fock glass") and the conductivity vanishes at a critical temperature proportional to the strength of disorder.