For a point source of light within the near field of circular, elliptical or tilted circular discs, diffraction into the geometrical shadow region is studied theoretically. For observation in the near field on the shadow side of the disc, the geometry of the caustic formed is shown to be closely approximated by the evolute of the diffracting edge, a result which holds strictly for plane wave incidence. For observation in the far field, the quite different patterns observed by Coulson and Becknell are explained using two approaches. First, a suitable approximation to the phase term in the Kirchhoff integral leads to a largely analytical treatment of the circular disc, and for the field on the axis of the elliptical disc. Second, the field away from the axis for the elliptical disc is explained by applying stationary phase methods to the Rubinowitz edge integral, in a modified form which is uniformly valid across the shadow boundary. Finally, the behaviour of wavefront dislocations formed in the diffracted field is studied.