The boundary of the convex hull of a compact algebraic curve in real 3-space defines a real algebraic surface. For general curves, that boundary surface is reducible, consisting of tritangent planes and stationary bisecants. We express the degree of this surface in terms of the degree, genus and singularities of the curve. We present methods for computing their defining polynomials, we show many colorful pictures, and we discuss extension to higher dimensions. This is joint work with Kristian Ranestad.