The representation theory of non-commutative Banach algebras parallels ideas from ring theory. We begin with a study of irreducible modules and representations. We define the radical and characterize it. The Jacobson density theorem shows that irreducible modules are highly transitive. Then we cover some automatic continuity results of B. Johnson. Finally we establish Cohen’s factorization theorem, which shows that a Banach algebra with a bounded approximate identity satisfies \({\mathfrak {A}}^2={\mathfrak {A}}\) .