  In the first part of the paper, 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, and when our algebra
is self-induced. In the second part of the paper, we mainly concentrate upon
dual Banach algebras. 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.
