PolySAT is a word-level decision procedure supporting bit-precise SMT reasoning over polynomial arithmetic with large bit-vector operations. Addressing challenges of verified software, PolySAT integrates the theoretical development of SMT-based calculi with a proof of concept implementation and empirical evaluation. The PolySAT calculus extends conflict-driven clause learning modulo theories with two key components: (i) a bit-vector plugin to the equality graph, and (ii) a theory solver for bit-vector arithmetic with non-linear polynomials. PolySAT implements dedicated procedures to extract bit-vector intervals from polynomial inequalities. For conflict analysis and resolution, PolySAT comes with on-demand lemma generation over non-linear bit-vector arithmetic. PolySAT is integrated into the SMT solver Z3 and has applications in model checking and smart contract verification where bit-blasting techniques on multipliers/divisions do not scale.