Active solids use elastic coupling between energy-consuming elements to achieve functionality inaccessible at equilibrium. For two-dimensional active surfaces, powerful experimental techniques allow for exquisite control over spatial patterning and fuel delivery. However, achieving such control in three-dimensional objects presents a challenge. I will present a continuum theory that describes an active surface wrapped around a passive soft solid. The competition between active surface stresses and bulk elasticity leads to a broad range of previously unexplored phenomena, which we have dubbed active elastocapillarity. In passive materials, positive surface tension rounds out corners and drives every shape towards a sphere. By contrast, activity can send the surface tension negative, which allows us to tune the target shape using elasticity. We discover that in these reconfigurable objects, material nonlinearity controls reversible switching and snap-through transitions between anisotropic shapes. Even for stable surfaces, a signature of activity arises in the negative group velocity of surface Rayleigh waves. These phenomena offer insights into living cellular membranes and underpin novel design principles across scales from robotic metamaterials down to shape-shifting nanoparticles.