The general quadratic Radon transform in two dimensions is investigated. Whereas the classical Radon transform of a smooth function represents the integration over all lines, the general quadratic Radon transform integrates over all conic sections. First, the parabolic isofocal Radon transform, i.e. the restriction of the general quadratic Radon transform to all parabolae with focus in the origin, is defined and illustrated. We show its intense relation to the classical Radon transform, deduce a support theorem, formulate an extension of the support theorem and derive an inversion formula. The natural extension to a more general class of isofocal quadratic Radon transforms is outlined. We show how the general quadratic Radon transform can be derived from the integrals over all parabolae by solving the related Cauchy problem. Finally, we introduce an entirely geometrical definition of a generalized Radon transform, the oriented generalized Radon transform.