This paper presents polar coding schemes for the two-user discrete memoryless broadcast channel (DM-BC) which achieve Marton's region with both common and private messages. This is the best achievable rate region known to date, and it is tight for all classes of two-user DM-BCs whose capacity regions are known. To accomplish this task, we first construct polar codes for both the superposition as well as binning strategy. By combining these two schemes, we obtain Marton's region with private messages only. Finally, we show how to handle the case of common information. The proposed coding schemes possess the usual advantages of polar codes, i.e., they have low encoding and decoding complexity and a superpolynomial decay rate of the error probability. We follow the lead of Goela, Abbe, and Gastpar, who recently introduced polar codes emulating the superposition and binning schemes. To align the polar indices, for both schemes, their solution involves some degradedness constraints that are assumed to hold between the auxiliary random variables and channel outputs. To remove these constraints, we consider the transmission of k blocks and employ a chaining construction that guarantees the proper alignment of the polarized indices. The techniques described in this paper are quite general, and they can be adopted to many other multiterminal scenarios whenever there polar indices need to be aligned.