The Hales-Jewett Theorem states that given any finite colouring
of the free semigroup S generated by a finite alphabet A, there is a
variable word over A all of whose instances are the same colour. We will
discuss various extensions and ramifications of the Hales-Jewett Theorem
including the Graham-Rothschild Parameter Sets Theorem. We present new
simple algebraic proofs of these results which rely on the algebraic
structure of the Stone-Čech compactification of the discrete semigroup
S. This is based on joint ongoing work with Neil Hindman and Dona Strauss.