This paper studies the concentration properties of random codes. Specifically, we show that, for discrete memoryless channels, the error exponent of a randomly generated code with pairwise-independent codewords converges in probability to its expectation -- the typical error exponent. For high rates, the result is a consequence of the fact that the random-coding error exponent and the sphere-packing error exponent coincide. For low rates, instead, the convergence is based on the fact that the union bound accurately characterizes the probability of error. The paper also zooms into the behavior at asymptotically low rates and shows that the error exponent converges in distribution to a Gaussian-like distribution. Finally, we present several results on the convergence of the error probability and error exponent for generic ensembles and channels.