A subset $S$ of an integral domain $R$ is called a semidomain provided that the pairs $(S,+)$ and $(S,\cdot)$ are semigroups with identities. The study of factorizations in integral domains was initiated by Anderson, Anderson, and Zafrullah in 1990, and this are has been systematically investigated since then. We study the divisibility and arithmetic of factorizations in the more general context of semidomains. We are especially concerned with the ascent of the most standard divisibility and factorization properties from a semidomain to its semidomain of (Laurent) polynomials. This is joint work with Felix Gotti.