In many important problems of supersonic flow, either for the whole field of flow or a part of it, the velocity components are constant on straight lines through a fixed point. Such velocity fields are called conical fields. In the usual theory of the linearised perturbations of a steady supersonic flow, the velocity is assumed to differ only slightly from a uniform undisturbed velocity, and in the defining equations and boundary conditions all non-linear terms in the components of the perturbation velocity (and their space derivatives) are neglected. In this paper the equations of linearised supersonic conical fields, and their general solution, are set out both for the region inside and for the region outside the Mach cone of the origin in the conical field. The results are applied to flow past plane triangular aerofoils with straight edges downstream of the vertex (of which there are six cases), to flow past those plane aerofoils of more extended shape and finite span for which the solution may be obtained by combining a finite number of conical fields, and to the problem of plane triangular vanes in semi-infinite free jets. In the applications, the velocity fields are determined in considerable detail, but a main purpose in setting them out was to exhibit the mathematical methods used and the physical considerations that enter in determining the mathematical solutions.