  Specht modules for an Ariki-Koike algebra have been investigated recently in
the context of cellular algebras. Thus, these modules are defined as quotient
modules of certain ``permutation'' modules, that is, defined as ``cell
modules'' via cellular bases. We shall introduce in this paper Specht modules
for an Ariki-Koike algebra as submodules of those ``permutation'' modules and
investigate their basic properties such as Standard Basis Theorem and the
ordinary Branching Theorem, generalizing several classical constructions for
type $A$. The second part of the paper moves on looking for Kleshchev's
branching rules for Specht and irreducible modules over an Ariki-Koike algebra.
We shall restrict to the case where the Ariki-Koike algebra has a semi-simple
bottom. With a recent work of Ariki on the classification of irreducible
modules, we conjecture that the results should be true in general if
$l$-regular multipartitions are replaced by Kleshchev's multipartitions. We
point out that our approach is independent of the use of cellular bases.
