  Hele-Shaw flow at vanishing surface tension is ill-defined. In finite time,
the flow develops cusp-like singularities. We show that the ill-defined problem
admits a weak {\it dispersive} solution when singularities give rise to a graph
of shock waves propagating in the viscous fluid. The graph of shocks grows and
branches. Velocity and pressure jump across the shock. We formulate a few
simple physical principles which single out the dispersive solution and
interpret shocks as lines of decompressed fluid. We also formulate the
dispersive weak solution in algebro-geometrical terms as an evolution of the
Krichever-Boutroux complex curve. We study in detail the most generic (2,3)
cusp singularity, which gives rise to an elementary branching event. This
solution is self-similar and expressed in terms of elliptic functions.
