  We survey the theory of Hopf monads on monoidal categories, and present new
examples and applications. As applications, we utilise this machinery to
present a new theory of cross products, as well as analogues of the Fundamental
Theorem of Hopf algebras and Radford's biproduct Theorem for Hopf algebroids.
Additionally, we describe new examples of Hopf monads which arise from Galois
and Ore extensions of bialgebras. We also classify Lawvere theories whose
corresponding monads on the category of sets and functions become Hopf, as well
as Hopf monads on the poset of natural numbers.
