ABSTRACT
A non-singular conic of the projective plane over the Galois field consists of q + 1 points no three of which are collinear. It is natural to ask if this non-collinearity condition for q + 1 points is sufficient for them to be a conic. There is a close relationship between arcs and certain algebraic curves, and between ovoids and circle geometries, projective planes and designs. Arcs and caps can be generalised by replacing their points with n-dimensional subspaces to obtain generalised k-arcs and generalised k-caps. These have strong connections to generalised quadrangles, projective planes, circle geometries, strongly regular graphs, and linear projective two-weight codes. In this survey results and problems concerning these objects will be mentioned. The focus will be on generalised ovals, generalised hyperovals, and generalised ovoids. These objects have strong connections to generalised quadrangles, projective planes, circle geometries, strongly regular graphs, linear projective two-weight codes, flocks of quadrics, ovoids of polar spaces, and other structure.
