Most theories having physical scales involved (namely, almost all the theories in physics) are actually *not* conformally invariant, because, to start with, they cannot be scale invariant.

and where are they encountered ?

Classical mechanics, fluid dynamics, quantum mechanics and quantum field theory are all not conformally invariant, at least in their full generalisation (low dimensional massless bosons quantum field theories are, instead).

In which areas of Physics are they a big hurdle ?

Not sure what you really mean by that. All the above are, to most extends, fully investigated. Every field theory is expected to have a symmetry group, namely the action is supposed to be invariant under some set of transformations (although it needs not necessarily be so). Once the symmetry group is specified, standard tools like, for instance, the Noether theorem, provide insights on conserved quantities that might help solving the equations of motion by simplifying some particular features. Large symmetry groups very often result in pretty *rigid* theories, where rigid means that there are almost no degrees of freedom for the dynamics (see also topological field theories).