We give necessary and sufficient conditions for sequences in the space AP(R) of continuous almost periodic functions on the real line to converge in the weak topology. The abstract results are illustrated by a number of examples which show that weak convergence seems to be a rare phenomenon. We also characterize the weakly compact subsets in AP(R). In particular, earlier statements made in the monograph by Dunford and Schwartz are refined and completed. We close with some open problems.