The aim of this paper is to report on the existence of a wide variety of exact solutions, ranging from black holes to wormholes, when a conformally coupled scalar field with a self-interacting potential containing a linear, a cubic and a quartic self interaction is taken as a source of the energy-momentum tensor, in the Einstein theory with a cosmological constant. Among all the solutions there are two particularly interesting. On the one hand, the spherically symmetric black holes when the cosmological constant is positive; they are shown to be everywhere regular, namely, there is no singularity neither inside nor outside the event horizon. On the other hand, there are spherically symmetric and topological wormholes that connect two asymptotically (anti) de Sitter regions with a different value for the cosmological constant. The regular black holes and the wormholes are supported by everywhere regular scalar field configurations.