We study the behavior of the Cr2+xSe alloys based on state-of-the-art first-principles electronic structure calculations. We show that these alloys are of special interest since they combine possible applications in spintronics devices with a series of diverse magnetic phenomena. First, we show that Cr2Se prefers the C1b structure while Cr3Se crystallizes in the D03 lattice. Our calculations suggest that as we dope Cr2Se with Cr atoms and move towards Cr3Se, all alloys are half-metallic fully compensated ferrimagnets (also known as half-metallic antiferromagnets) with a gap in the spin-down band. All alloys follow a generalized version of the Slater-Pauling rule for the Heusler compounds and we show that for Cr3Se a small deviation occurs due to the antibonding single band created by the 4s states of the Cr and Se atoms in the spin-down band structure which crosses the Fermi level. In the case of Cr3Se we observe a metamagnetic behavior under hydrostatic pressure since Cr atoms with different symmetry present both itinerant and localized magnetic properties. Finally, calculations based on the frozen-magnon approximation reveal that the strong intersublattice antiferromagnetic coupling between the nearest-neighboring Cr atoms stabilizes the ferrimagnetic character of both Cr2Se and Cr3Se and leads to estimated Curie temperature exceeding considerably the room temperature. Combination of this feature together with the half-metallic antiferromagnetism makes the Cr2+xSe alloys ideal for realistic spintronics applications.