Configuration space (C-space) has played a central role in collision-free motion planning, particularly for robot manipulators. While it is possible to check for collisions at a point using standard algorithms, to date no practical method exists for computing collision-free C-space regions with rigorous certificates due to the complexities of mapping task-space obstacles through the kinematics. In this work, we present the first to our knowledge method for generating such regions and certificates through convex optimization. Our method, called C-Iris (C-space Iterative Regional Inflation by Semidefinite programming), generates large, convex polytopes in a rational parametrization of the configuration space which are guaranteed to be collision-free. Such regions have been shown to be useful for both optimization-based and randomized motion planning. Our regions are generated by alternating between two convex optimization problems: (1) a simultaneous search for a maximal-volume ellipse inscribed in a given polytope and a certificate that the polytope is collision-free and (2) a maximal expansion of the polytope away from the ellipse which does not violate the certificate. The volume of the ellipse and size of the polytope are allowed to grow over several iterations while being collision-free by construction. Our method works in arbitrary dimensions, only makes assumptions about the convexity of the obstacles in the task space, and scales to realistic problems in manipulation. We demonstrate our algorithm's ability to fill a non-trivial amount of collision-free C-space in a 3-DOF example where the C-space can be visualized, as well as the scalability of our algorithm on a 7-DOF KUKA iiwa and a 12-DOF bimanual manipulator.