The paper deals with the space S1-3- of piecewise cubic C¹ functions defined on an arbitrary triangulation. This space has the distinction (among all C¹ spline spaces) of offering the smallest possible polynomial degree that, in general, supplies more parameters than there are vertices in the triangulation. However, at present its dimension is unknown, and it is also unknown whether, in general, the interpolation conditions at the vertices and the smoothness conditions across interior edges are consistent. The author acknowledges these problems but does not otherwise address them. Instead, he reviews a well-known lower bound on the dimension, describes four ways of representing splines in S1-3-, derives equations that enforce certain boundary conditions, and gives some numerical results for least squares approxi- mation and interpolation to scattered data. There is a useful bibliography of 38 items.