A recurrent problem in materials science is the prediction of the percolation threshold of suspensions and composites containing complex-shaped constituents. We consider an idealized material built up from freely overlapping objects randomly placed in a matrix, and numerically compute the geometrical percolation threshold ${\mathit{p}}_{\mathit{c}}$ where the objects first form a continuous phase. Ellipsoids of revolution, ranging from the extreme oblate limit of platelike particles to the extreme prolate limit of needlelike particles, are used to study the influence of object shape on the value of ${\mathit{p}}_{\mathit{c}}$. The reciprocal threshold 1/${\mathit{p}}_{\mathit{c}}$ (${\mathit{p}}_{\mathit{c}}$ equals the critical volume fraction occupied by the overlapping ellipsoids) is found to scale linearly with the ratio of the larger ellipsoid dimension to the smaller dimension in both the needle and plate limits. Ratios of the estimates of ${\mathit{p}}_{\mathit{c}}$ are taken with other important functionals of object shape (surface area, mean radius of curvature, radius of gyration, electrostatic capacity, excluded volume, and intrinsic conductivity) in an attempt to obtain a universal description of ${\mathit{p}}_{\mathit{c}}$. Unfortunately, none of the possibilities considered proves to be invariant over the entire shape range, so that ${\mathit{p}}_{\mathit{c}}$ appears to be a rather unique functional of object shape. It is conjectured, based on the numerical evidence, that 1/${\mathit{p}}_{\mathit{c}}$ is minimal for a sphere of all objects having a finite volume.