Transport properties of 2D superconducting systems can be very different from those of bulk superconductors because thermal and quantum fluctuations of superconducting order parameter are more pronounced and play a crucial role. First we focus on the influence of superconducting fluctuations on dynamics, while the system is in the normal state but close to the superconducting transition. In the fluctuational regime, we derive a Ginzburg-Landau-type action under far-from equilibrium conditions. Then, utilizing it, we calculate the fluctuation-induced density of states and Maki-Thomson- and Aslamazov-Larkin-type contributions to the in-plane electrical conductivity [1,2]. We propose an experimental setup where our results can be tested: thin superconducting film sandwiched between a gate and a substrate, which have different temperatures and different electrochemical potentials. Then, we concentrate on transport at lower temperatures in close-to-equilibrium conditions investigating influence of quantum fluctuations on the unbinding of vortex-antivortex pairs. We determine the temperature below which quantum fluctuations dominate over thermal fluctuations and describe the transport in this quantum regime. The crossover from the quantum to the classical regime is discussed and the quantum correction to the classical current-voltage relation is determined [3]. [1] Phys. Rev. Lett. 105, 187003 (2010) [2] Phys. Rev. B 84, 064510 (2011) [3] Phys. Rev. B 80, 212504 (2009)