A wireless massive MIMO system entails a large number (tens or hundreds) of base station antennas serving a much smaller number of users, with large gains in spectral-efficiency and energy-efficiency compared with conventional MIMO technology. Until recently it was believed that in multi-cellular massive MIMO system, even in the asymptotic regime, as the number of service antennas tends to infinity, the performance is limited by directed inter-cellular interference. This interference results from unavoidable re-use of reverse-link training sequences (pilot contamination) by users in different cells. We devise a new concept that leads to the effective elimination of inter-cell interference in massive MIMO systems. This is achieved by outer multi-cellular precoding, which we call Large-Scale Fading Precoding (LSFP). The main idea of LSFP is that each base station linearly combines messages aimed to users from different cells that re-use the same training sequence. Crucially, the combining coefficients depend only on the slow-fading coefficients between the users and the base stations. Each base station independently transmits its LSFP-combined symbols using conventional linear precoding that is based on estimated fast-fading coefficients. Further, we derive estimates for downlink and uplink SINRs and capacity lower bounds for the case of massive MIMO systems with LSFP and a finite number of base station antennas.