  In Volborthite, spin-1/2 moments form a distorted Kagom\'e lattice, of corner
sharing isosceles triangles with exchange constants $J$ on two bonds and $J'$
on the third bond. We study the properties of such spin systems, and show that
despite the distortion, the lattice retains a great deal of frustration.
Although sub-extensive, the classical ground state degeneracy remains very
large, growing exponentially with the system perimeter. We consider degeneracy
lifting by thermal and quantum fluctuations. To linear (spin wave) order, the
degeneracy is found to stay intact. Two complementary approaches are therefore
introduced, appropriate to low and high temperatures, which point to the same
ordered pattern. In the low temperature limit, an effective chirality
Hamiltonian is derived from non-linear spin waves which predicts a transition
on increasing $J'/J$, from $\sqrt 3\times \sqrt 3$ type order to a new
ferrimagnetic {\em striped chirality} order with a doubled unit cell. This is
confirmed by a large-N approximation on the O($n$) model on this lattice. While
the saddle point solution produces a line degeneracy, $O(1/n)$ corrections
select the non-trivial wavevector of the striped chirality state. The quantum
limit of spin 1/2 on this lattice is studied via exact small system
diagonalization and compare well with experimental results at intermediate
temperatures. We suggest that the very low temperature spin frozen state seen
in NMR experiments may be related to the disconnected nature of classical
ground states on this lattice, which leads to a prediction for NMR line shapes.
