  Twin-width is a graph width parameter recently introduced by Bonnet, Kim,
Thomass\'{e} & Watrigant. Given two graphs $G$ and $H$ and a graph product
$\star$, we address the question: is the twin-width of $G\star H$ bounded by a
function of the twin-widths of $G$ and $H$ and their maximum degrees? It is
known that a bound of this type holds for strong products (Bonnet, Geniet, Kim,
Thomass\'{e} & Watrigant; SODA 2021). We show that bounds of the same form hold
for Cartesian, tensor/direct, corona, rooted, replacement, and zig-zag
products. For the lexicographical product it is known that the twin-width of
the product of two graphs is exactly the maximum of the twin-widths of the
individual graphs (Bonnet, Kim, Reinald, Thomass\'{e} & Watrigant; IPEC 2021).
In contrast, for the modular product we show that no bound can hold. In
addition, we provide examples showing many of our bounds are tight, and give
improved bounds for certain classes of graphs.
