  A Hopf monoid (in Joyal's category of species) is an algebraic structure akin
to that of a Hopf algebra. We provide a self-contained introduction to the
theory of Hopf monoids in the category of species. Combinatorial structures
which compose and decompose give rise to Hopf monoids. We study several
examples of this nature. We emphasize the central role played in the theory by
the Tits algebra of set compositions. Its product is tightly knit with the Hopf
monoid axioms, and its elements constitute universal operations on connected
Hopf monoids. We study analogues of the classical Eulerian and Dynkin
idempotents and discuss the Poincare-Birkhoff-Witt and Cartier-Milnor-Moore
theorems for Hopf monoids.
