A class of associative algebras (''cellular'') is defined by means of
multiplicative properties of a basis. They are shown to have cell repr
esentations whose structure depends on certain invariant bilinear form
s. One thus obtains a general description of their irreducible represe
ntations and block theory as well as criteria for semisimplicity. Thes
e concepts are used to discuss the Brauer centraliser algebras, whose
irreducibles are described in full generality, the Ariki-Koike algebra
s, which include the Hecke algebras of type A and B and (a generalisat
ion of) the Temperley-Lieb and Jones' recently defined ''annular'' alg
ebras. In particular the latter are shown to be non-semisimple when th
e defining parameter delta satisfies gamma(g(n))(-delta/2)=1, where ga
mma(n), is the n-th Tchebychev polynomial and g(n) is a quadratic poly
nomial.