FASER is a new experiment at the Large Hadron Collider (LHC) aiming to search for light, weakly interacting new particles, complementing other experiments. The particle detector will be located 480 m downstream of the ATLAS interaction point. In addition to searches for new particles, we also aim to study high-energy neutrinos of all flavors, as there is a huge flux of neutrinos at this location. To date, muon-neutrino cross-section data, obtained with accelerator-based neutrino beams, exist up to 350 GeV, but we are still missing data at the TeV energy scale. At LHC-FASER, the neutrino cross sections will be measured in the currently unexplored energy range between 350 GeV and 6 TeV. In particular, tau neutrinos will be measured at the highest energy ever. Furthermore, the channels associated with heavy quark (charm and beauty) production can be studied. As a feasibility study, we performed a test run in 2018 at the proposed detector location, using a 30 kg lead/tungsten emulsion-based neutrino detector. We collected 12.5 fb$^{-1}$ of data, and expect a few tens of neutrino interactions to have been recorded in the detector. From 2021 to 2023 during Run 3 of the 14 TeV LHC, we are planning to deploy an emulsion detector with a target mass of 1.2 tons, possibly coupled with the FASER magnetic spectrometer, which would yield roughly 1,300 $\nu_e$, 20,000 $\nu_{\mu}$, and 20 $\nu_{\tau}$ interacting in the detector. Here, we present an overview of the FASER neutrino program, as well as a preliminary analysis of the 2018 test run.