  We introduce a simple combinatorial way, which we call a rectangular diagram
of a surface, to represent a surface in the three-sphere. It has a particularly
nice relation to the standard contact structure on $\mathbb S^3$ and to
rectangular diagrams of links. By using rectangular diagrams of surfaces we are
going, in particular, to develop a method to distinguish Legendrian knots. This
requires a lot of technical work of which the present paper addresses only the
first basic question: which isotopy classes of surfaces can be represented by a
rectangular diagram. Vaguely speaking the answer is this: there is no
restriction on the isotopy class of the surface, but there is a restriction on
the rectangular diagram of the boundary link that can arise from the
presentation of the surface. The result extends to Giroux's convex surfaces for
which this restriction on the boundary has a natural meaning. In a subsequent
paper we are going to consider transformations of rectangular diagrams of
surfaces and to study their properties. By using the formalism of rectangular
diagrams of surfaces we also produce here an annulus in $\mathbb S^3$ that we
expect to be a counterexample to the following conjecture: if two Legendrian
knots cobound an annulus and have zero Thurston--Bennequin numbers relative to
this annulus, then they are Legendrian isotopic.
