In this paper, we propose a novel joint caching and massive multiple-input multiple-output (MIMO) transmission scheme, referred to as \emph{cache-aided massive MIMO}, for multi-cell downlink transmission to multiple cache-enabled receivers. With the proposed scheme, users who have cached (a portion of) the files that they request are offloaded and, hence, (partially) inactive during downlink transmission. The other users either benefit from the cache-enabled offloading for mitigating pilot contamination or exploit the cached but unrequested files to cancel interference during uplink channel estimation and downlink file reception. Moreover, by redesigning the transmit precoders based on the cache status of the users and channel state information, we gain additional degrees of freedom for massive MIMO transmission. For a given cache status, we analyze the equivalent content delivery rates (ECDRs), i.e., the average rates of delivering a requested file via both caching and massive MIMO transmission to the requesting user, for cache-aided massive MIMO employing re-designed maximum ratio transmission (MRT), zero-forcing (ZF) precoding, and regularized zero-forcing (RZF) precoding. Based on the derived results, the impact of (random) uncoded caching and coded caching on the performance of the re-designed precoding schemes is investigated. Simulation results validate our derivations and show that caching is beneficial for precoded downlink transmission as it enhances the transmit power allocation, mitigates intra- and inter-cell interference, and reduces the impairment caused by pilot contamination. Compared with conventional massive MIMO without caching and with cache-oblivious precoding, the proposed cache-aided massive MIMO scheme achieves a significantly higher ECDR even when the number of users approaches the number of transmit antennas.