The 21-cm probability distribution function (PDF; i.e. distribution of pixel brightness temperatures) is expected to be highly non-Gaussian during reionization and to provide important information on the distribution of density and ionization. We measure the 21-cm PDF as a function of redshift in a large simulation of cosmic reionization and propose a simple empirical fit. Guided by the simulated PDF, we then carry out a maximum likelihood analysis of the ability of upcoming experiments to measure the shape of the 21-cm PDF and derive from it the cosmic reionization history. Under the strongest assumptions, we find that upcoming experiments can measure the reionization history in the mid to late stages of reionization to 1-10 per cent accuracy. Under a more flexible approach that allows for four free parameters at each redshift, a similar accuracy requires the lower noise levels of second-generation 21-cm experiments.