This paper examines what is now called the Saffman-Taylor instability, describing the stability of the interface between two fluids flowing through a porous medium when one of lesser viscosity is forced into one of greater viscosity. The instability results in the growth of ``fingers'' of the less viscous fluid penetrating the more viscous fluid. Development of such an instability is unwanted when extracting oil from porous rock: if the speed of extraction is too great then water fingers penetrate the oil. This instability is also relevant to problems such as directional solidification, dendritic formation and electro-deposition.