  The Hamilton--Jacobi formalism generalized to 2--dimensional field theories
according to Lepage's canonical framework is applied to several covariant real
scalar fields, e.g. massless and massive Klein--Gordon, Sine--Gordon, Liouville
and $\phi^4$ theories. For simplicity we use the Hamilton--Jacobi equation of
DeDonder and Weyl. Unlike mechanics we have to impose certain integrability
conditions on the velocity fields to guarantee the transversality relations
between Hamilton--Jacobi wave fronts and the corresponding families of
extremals embedded therein. B\"acklund Transformations play a crucial role in
solving the resulting system of coupled nonlinear PDEs.
