{\em Quasi-quadrics} were introduced by Penttila, De Clerck, O'Keefe and Hamilton in [2]. They are defined as point sets which have the same intersection numbers with respect to hyperplanes as non-singular quadrics. We extend this definition in two ways.
The first extension is to {\em quasi-Hermitian varieties}, which are point sets which have the same intersection numbers with respect to hyperplanes as non-singular Hermitian varieties.
The second one is to {\em singular quasi-quadrics}, i.e. point sets $\mathcal{K}$ which have the same intersection numbers with respect to hyperplanes as singular quadrics. Our starting point was to investigate whether every singular quasi-quadric is a cone over a non-singular quasi-quadric. This question is tackled in the case of a point set $\mathcal{K}$ with the same intersection numbers with respect to hyperplanes as a point over an ovoid.