We present a short survey of the theory of multipliers, or double
centralisers, of Banach algebras and completely contractive Banach algebras.
Our approach is very algebraic: this is a deliberate attempt to separate
essentially algebraic arguments from topological arguments. We concentrate upon
the problem of how to extend module actions, and homomorphisms, from algebras
to multiplier algebras. We then consider the special cases when we have a
bounded approximate identity, when our algebra is self-induced, and when we
have a dual Banach algebra. We provide a simple criterion for when a multiplier
algebra is a dual Banach algebra. This is applied to show that the multiplier
algebra of the convolution algebra of a locally compact quantum group is always
a dual Banach algebra. We also study this problem within the framework of
abstract Pontryagin duality, and show that we construct the same
weak$^*$-topology. We explore the notion of a Hopf convolution algebra, and
show that in many cases, the use of the extended Haagerup tensor product can be
replaced by a multiplier algebra.