  We show that the general theory of relativity can be formulated in the
language of Weyl geometry. We develop the concept of Weyl frames and point out
that the new mathematical formalism may lead to different pictures of the same
gravitational phenomena. We show that in an arbitrary Weyl frame general
relativity, which takes the form of a scalar-tensor gravitational theory, is
invariant with respect to Weyl tranformations. A kew point in the development
of the formalism is to build an action that is manifestly invariant with
respect to Weyl transformations. When this action is expressed in terms of
Riemannian geometry we find that the theory has some similarities with
Brans-Dicke gravitational theory. In this scenario, the gravitational field is
not described by the metric tensor only, but by a combination of both the
metric and a geometrical scalar field. We illustrate this point by, firstly,
discussing the Newtonian limit in an arbitrary frame, and, secondly, by
examining how distinct geometrical and physical pictures of the same phenomena
may arise in different frames. To give an example, we discuss the gravitational
spectral shift as viewed in a general Weyl frame. We further explore the
analogy of general relativity with scalar-tensor theories and show how a known
Brans-Dicke vacuum solution may appear as a solution of general relativity
theory when reinterpreted in a particular Weyl frame. Finally, we show that the
so-called WIST gravity theories are mathematically equivalent to Brans-Dicke
theory when viewed in a particular frame.
