Fluid antenna is a novel technology that can make an antenna appear instantly at one of N preset locations in a predefined space. An important application is to adopt fluid antenna in a small space of mobile device for obtaining the tremendous diversity hidden in the small space. Previous results have revealed that a single-antenna fluid antenna system, even with a very small space, can outperform a multiple antenna maximum ratio combining (MRC) system if $N$ is large enough. This paper explores the potential of using fluid antenna for multiple access through performance analysis. Fluid antenna multiple access (FAMA) exploits moments of deep fade experienced by the interference to achieve a favourable channel condition for the desired signal, without requiring sophisticated signal processing. We analyze the FAMA system 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 capacity 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 N is required to achieve a given multiplexing gain using fluid antennas with a given size. Results illustrate that it is possible for FAMA to support hundreds of users using only one fluid antenna at each user in a few wavelengths of space, giving rise to significant enhancement in the network outage capacity.