Building on polar code constructions proposed by the authors for deterministic broadcast channels, two theorems are introduced in the present paper for noisy two-user broadcast channels. The theorems establish polar code constructions for two important information-theoretic broadcast strategies: (1) Cover's superposition strategy; (2) Marton's construction. One aspect of the polar code constructions is the alignment of polarization indices via constraints placed on the auxiliary and channel-input distributions. The codes achieve capacity-optimal rates for several classes of broadcast channels (e.g., binary-input stochastically degraded channels). Applying Arkan's original matrix kernel for polarization, it is shown that the average probability of error in decoding two private messages at the broadcast receivers decays as O(2 -nβ) where 0 < β < 1/2 and n is the code length. The encoding and decoding complexities remain O(n log n). The error analysis is made possible by defining new polar code ensembles for broadcast channels.