  Although it is important both in theory as well as in applications, a theory
of Birkhoff interpolation with main emphasis on the shape of the set of nodes
is still missing. Although we will consider various shapes (e.g. we find all
the shapes for which the associated Lagrange problem has unique solution), we
concentrate on one of the simplest shapes:``rectangular'' (also called
"cartesian grids"). The ultimate goal is to obtain a geometrical understanding
of the solvability. We partially achieve this by describing several regularity
criteria, which we illustrate by many examples. At the end we discuss several
conjectures which, we think, are important in understanding the behaviour of
Birkhoff interpolation schemes in higer dimensions. Although we prove these
conjectures in many unrelated cases, we believe that a ``complete proof''
requires new ideas which go beyond the usual methods in interpolation theory
(and may reach areas such as algebraic geometry or algebraic topology).
