  The \emph{equations of Boussinesq approximation} (EBA) for an incompressible
and inhomogeneous in density fluid are analyzed from a viewpoint of the
asymptotic theory. A systematic scaling shows that there is an infinite number
of related asymptotic models. We have divided them into three classes: `poor',
`reasonable' and `good' Boussinesq approximations. Each model can be
characterized by two parameters $q$ and $k$, where $q =1, 2, 3, \dots$ and
$k=0, \pm 1, \pm 2,\dots$. Parameter $q$ is related to the `quality' of
approximation, while $k$ gives us an infinite set of possible scales of
velocity, time, viscosity, \emph{etc.} Increasing $q$ improves the quality of a
model, but narrows the limits of its applicability. Parameter $k$ allows us to
vary the scales of time, velocity and viscosity and gives us the possibility to
consider any initial and boundary conditions. In general, we discover and
classify a rich variety of possibilities and restrictions, which are hidden
behind the routine use of the Boussinesq approximation. The paper is devoted to
the multiplicity of scalings and related restrictions. We do not study any
particular solutions and particular failures of EBA.
