This paper extends the theory of false discovery rates (FDR) pioneered by Benjamini and Hochberg (1995). We give statistical models underlying multiple testing in the independent, continuous case. We define the realized FDR and False Nondiscovery Rate (FNR) as stochastic processes and characterize their asymptotic behavior. We develop methods for estimating the p-value distribution, even in the non-identifiable case. We study a class of methods that asymptotically attains the optimal behavior. We also develop a new method for controlling the probability of large realized FDR.