Fluid antenna system represents an emerging technology that enables an antenna to switch its physical location in a predefined space. This paper explores the potential of using a single fluid antenna at each mobile user for multiple access, which we refer to it as fluid antenna multiple access (FAMA). FAMA exploits spatial moments of deep fade suffered by the interference to achieve a favourable channel condition for the desired signal, without requiring sophisticated signal processing. We analyze the FAMA network by first deriving the outage probability of the signal-to-interference ratio (SIR) in a double integral form. We then obtain an outage probability upper bound in closed form and an average outage rate lower bound for the FAMA system, with an arbitrary number of interferers, from which the multiplexing gain of FAMA is characterized. We also estimate how large the number of locations is required to achieve a given multiplexing gain using fluid antennas with a given size. Results show that it is possible for FAMA to support hundreds of users using only one fluid antenna of a few wavelengths of space at each user, giving rise to significant gain in the average network outage rate.