It seems to me like every book on representation theory leaps into groups right away, even though the underlying ideas, such as representations, convolution algebras, etc. don't really make explicit use of inverses.  This leads to the fairly natural question of how much of representation theory still works for monoids (or even semigroups)?  

I would imagine that irreducible representations probably exist in some form, since all you really need is to have some way to reduce monoid-modules over the complex numbers.  Of course it could also be that monoid representation theory is in some form fatally flawed and reduces down to something boring for reasons that are not obvious to me at this point.

Certainly irreducible representations exist; one can still construct the monoid algebra of a monoid and consider modules over the algebra.  But Maschke's theorem is false in general for finite monoids.  Indeed, consider the monoid $M = \langle x | x^3 = x^2 \rangle$.  Complex (for the sake of argument) representations of $M$ are the same as representations of the monoid algebra $\mathbb{C}[x]/(x^3 - x^2)$ and this algebra is not semisimple, so its finite-dimensional representations are not completely reducible.  (Note that the usual proof of Maschke's theorem fails miserably; you can't average inner products over a monoid without inverses, and any unitary representation of a monoid has to factor through the Grothendieck group).

So part of the answer may just be that the representation theory of monoids is inherently more complicated.  Although this isn't doesn't seem to be stressed much in textbooks, having inverses is a pretty important structural property of groups; it endows group algebras with an antipode and endows the category of representations of a group with duals.  
