In this talk I will present joint work with F. Schulze on the short-time existence of the Ricci flow from initial data with isolated conical singularities. Assuming the links of the cones that model the singularities are spheres with curvature operator greater that one, we construct a smooth Ricci flow which achieves the initial condition in the Gromov-Hausdorff sense and smoothly away from the singular points. On the other hand, if the links of the cones that model the singularities are quotients of such spheres, we construct a smooth solution to the orbifold Ricci flow. Moreover, the tangent flows at the singular points are asymptotically conical expanders with positive curvature operator or quotients of them respectively.