  We discuss some problems related to dimensional reductions of gravity
theories to two-dimensional and one-dimensional dilaton gravity models. We
first consider the most general cylindrical reductions of the four-dimensional
gravity and derive the corresponding (1+1)-dimensional dilaton gravity, paying
a special attention to a possibility of producing nontrivial cosmological
potentials from pure geometric variables (so to speak, from `nothing'). Then we
discuss further reductions of two-dimensional theories to the dimension one by
a general procedure of separating the space and time variables. We illustrate
this by the example of the spherically reduced gravity coupled to scalar
matter. This procedure is more general than the usual `naive' reduction and
apparently more general than the reductions using group theoretical methods. We
also explain in more detail the earlier proposed `static-cosmological' duality
(SC-duality) and discuss some unusual cosmologies and static states which can
be obtained by using the method of separating the space and time variables.
