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Integral transforms which map functions on onto their integrals on circular cones having fixed axis direction and variable opening angle are introduced and studied as generalizations of the known Radon transform. Besides their intrinsic mathematical interest, they serve as backbone support to emission imaging based on Compton scattered radiation, the way the standard Radon transform does for emission imaging based on non-scattered radiation. In this work, we establish its basic properties and prove analytically its invertibility. Formulae to express it in terms of the standard Radon transform (or vice versa) are given. We also discuss some extensions as applications.