A theoretical account is given of experiments performed by Saffman & Taylor (1958) in which a fluid drives a liquid out of a long straight channel of very small thickness formed between two parallel sheets sealed at the edges. The penetrating fluid forms a long finger, whose sides are parallel to the edges of the channel, and which has a rounded tip, which advances with unaltered shape at a constant speed U. The theory correctly predicts the shape of the finger as a function of the ratio λ = (asymptotic width of finger)/(width of channel) and gives the relation between λ and U, which is in good agreement with experiment. In particular it shows that, as U increases from zero to infinity, λ steadily decreases from 1 to 0·5.