Low-complexity polar codes are introduced for discrete memoryless broadcast channels. For non-linear and linear $m$-user binary-output deterministic broadcast channels, polarization-based codes achieve optimal rates within the private-message capacity region. By contrast to random binning strategies requiring exponential complexity, polarization-based codes provide encoding and decoding complexities of $O(n*\log n)$ where $n$ is the block length, with a stretched-exponential decay of the average block error probability. For noisy broadcast channels with $m = 2$ users, low-complexity polar codes are developed according to: i) Cover's superposition strategy; ii) Marton's coding scheme. Experiments over finite code lengths corroborate the theory.