  By using the systematic approach of parent action method, we derive one
Weyl-noninvariant and two Weyl-invariant actions of bosonic $p$-branes ($p\geq
2$) starting from the Nambu-Goto action, and establish the duality symmetries
in this set of four actions. Moreover, we discover a new bosonic $p$-brane
action (including the string theory) and deduce two corresponding
Weyl-invariant formulations by proposing a new special parent action. We find
that the same duality symmetries as those mentioned above exist in this new set
of actions. The new $p$-brane actions are also briefly analyzed.
