Neutrino oscillations physics is entered in the precision era. In this context accelerator-based neutrino experiments need a reduction of systematic errors to the level of a few percent. Today one of the most important sources of systematic errors are neutrino-nucleus cross sections which in the hundreds-MeV to few-GeV energy region are known with a precision not exceeding 20%. In this article we review the present experimental and theoretical knowledge of the neutrino-nucleus interaction physics. After introducing neutrino oscillation physics and accelerator-based neutrino experiments, we overview general aspects of the neutrino-nucleus cross sections, both theoretical and experimental views. Then we focus on these quantities in different reaction channels. We start with the quasielastic and quasielastic-like cross section, putting a special emphasis on multinucleon emission channel which attracted a lot of attention in the last few years. We review the main aspects of the different microscopic models for this channel by discussing analogies and differences among them.The discussion is always driven by a comparison with the experimental data. We then consider the one pion production channel where data-theory agreement remains very unsatisfactory. We describe how to interpret pion data, then we analyze in particular the puzzle related to the impossibility of theoretical models and Monte Carlo to simultaneously describe MiniBooNE and MINERvA experimental results. Inclusive cross sections are also discussed, as well as the comparison between the $\nu_\mu$ and $\nu_e$ cross sections, relevant for the CP violation experiments. The impact of the nuclear effects on the reconstruction of neutrino energy and on the determination of the neutrino oscillation parameters is reviewed. A window to the future is finally opened by discussing projects and efforts in future detectors, beams, and analysis.