  The 3+1 formulation of scalar-tensor theories of gravity (STT) is obtained in
the physical (Jordan) frame departing from the 4+0 covariant field equations.
Contrary to the common belief (folklore), the new system of ADM-like equations
shows that the Cauchy problem of STT is well formulated (in the sense that the
whole system of evolution equations is of first order in the time-derivative).
This is the first step towards a full first order (in time and space)
formulation from which a subsequent hyperbolicity analysis (a well-posedness
determination) can be performed. Several gauge (lapse and shift) conditions are
considered and implemented for STT. In particular, a generalization of the
harmonic gauge for STT allows us to prove the well posedness of the STT using a
second order analysis which is very similar to the one used in general
relativity. Some spacetimes of astrophysical and cosmological interest are
considered as specific applications. Several appendices complement the ideas of
the main part of the paper.
