A spiderweb central configuration is a configuration in which the masses lie at the intersection points of a collection of concentric circles with another collection of lines, meeting at the center of the circles with equal angles. The intersection points of the lines with a single circle should all be occupied by points of equal mass, but the masses may vary from circle to circle. An additional mass (which may be zero) is placed at the center of the system. For any desired number of lines, number of circles, and profile of the masses on each concentric circle of a spiderweb central configuration, it is possible to find a spiderweb central configuration matching those parameters. One can similarly obtain central configurations for families of nested Platonic solids, or more generally group-theoretic orbits of any finite subgroup of the orthogonal group.