  Conformal invariant new forms of $p$-brane and D$p$-brane actions are
proposed. The field content of these actions are: an induced metric, gauge
fields, an auxiliary metric and an auxiliary scalar field that implements the
Weyl invariance. This scalar field transforms with a conformal weight that
depends on the brane dimension. The proposed actions are Weyl invariant in any
dimension and the elimination of the auxiliary metric and the scalar field
reproduces the Nambu-Goto action for $p$-branes and the Born-Infeld action for
D$p$-branes. As a consequence of the fact that in the $p$-brane case, the
action is quadratic in $\partial_0 X$, we develop a complete construction for
the associated canonical formalism, solving the problem in previous
formulations of conformal $p$-brane actions where the Hamiltonian can not be
constructed. We obtain the algebra of constraints and identify the generator of
the Weyl symmetry. In the construction of the corresponding supersymmetric
generalization of this conformal $p$-brane action we show that the associated
$\kappa$-symmetry is consistent with the conformal invariance. In the
D$p$-brane case, the actions are quadratic in the gauge fields. These actions
can be used to construct new conformal couplings in any dimension $p$ to the
auxiliary scalar field now promoted as a dynamical variable.
