We investigate topological Cooper pairing, including gapless Weyl and fully gapped class DIII superconductivity, in a three-dimensional doped Luttinger semimetal. The latter describes effective spin-3/2 carriers near a quadratic band touching and captures the normal-state properties of the 227 pyrochlore iridates and half-Heusler alloys. Electron-electron interactions may favor non- s -wave pairing in such systems, including even-parity d -wave pairing. We argue that the lowest energy d -wave pairings are always of complex (e.g., d + i d ) type, with nodal Weyl quasiparticles. This implies ϱ ( E ) ∼ | E | 2 scaling of the density of states (DoS) at low energies in the clean limit or ϱ ( E ) ∼ | E | over a wide critical region in the presence of disorder. The latter is consistent with the T dependence of the penetration depth in the half-Heusler compound YPtBi. We enumerate routes for experimental verification, including specific heat, thermal conductivity, NMR relaxation time, and topological Fermi arcs. Nucleation of any d -wave pairing also causes a small lattice distortion and induces an s -wave component; this gives a route to strain-engineer exotic s + d pairings. We also consider odd-parity, fully gapped p -wave superconductivity. For hole doping, a gapless Majorana fluid with cubic dispersion appears at the surface. We invent a generalized surface model with ν -fold dispersion to simulate a bulk with winding number ν . Using exact diagonalization, we show that disorder drives the surface into a critically delocalized phase, with universal DoS and multifractal scaling consistent with the conformal field theory (CFT) SO( n ) ν , where n → 0 counts replicas. This is contrary to the naive expectation of a surface thermal metal, and implies that the topology tunes the surface renormalization group to the CFT in the presence of disorder.