3D reconstruction from multiple views is a successful computer vision field with multiple deployments in applications. State of the art is based on traditional RGB frames that enable optimization of photo-consistency cross views. In this paper, we study the problem of 3D reconstruction from event-cameras, motivated by the advantages of event-based cameras in terms of low power and latency as well as by the biological evidence that eyes in nature capture the same data and still perceive well 3D shape. The foundation of our hypothesis that 3D reconstruction is feasible using events lies in the information contained in the occluding contours and in the continuous scene acquisition with events. We propose Apparent Contour Events (ACE), a novel event-based representation that defines the geometry of the apparent contour of an object. We represent ACE by a spatially and temporally continuous implicit function defined in the event x-y-t space. Furthermore, we design a novel continuous Voxel Carving algorithm enabled by the high temporal resolution of the Apparent Contour Events. To evaluate the performance of the method, we collect MOEC-3D, a 3D event dataset of a set of common real-world objects. We demonstrate the ability of EvAC3D to reconstruct high-fidelity mesh surfaces from real event sequences while allowing the refinement of the 3D reconstruction for each individual event.