  We define the Zappa-Sz\'{e}p product of a Fell bundle by a groupoid, which
turns out to be a Fell bundle over the Zappa-Sz\'{e}p product of the underlying
groupoids. Under certain assumptions, every Fell bundle over the Zappa-Sz\'{e}p
product of groupoids arises in this manner. We then study the representation
associated with the Zappa-Sz\'{e}p product Fell bundle and show its relation to
covariant representations. Finally, we study the associated universal
C*-algebra, which turns out to be a C*-blend, generalizing an earlier result
about the Zappa-Sz\'{e}p product of groupoid C*-algebras. In the case of
discrete groups, the universal C*-algebra of a Fell bundle embeds injectively
inside the universal C*-algebra of the Zappa-Sz\'{e}p product Fell bundle.
