Highly accurate relativistic Gaussian basis sets are developed for the 103 elements from H(Z=1) to Lr (Z=103). Orbital exponents are optimized by minimizing the atomic self-consistent field (SCF) energy with the scalar relativistic third-order Douglas-Kroll approximation. The basis sets are designed to have equal quality and to be appropriate for the incorporation of relativistic effects. The basis set performance is tested by calculations on prototypical molecules, hydrides, and dimers of copper, silver, and gold using SCF, Moller-Plesset theory, and the singles and doubles coupled-cluster methods with and without perturbative triples [CCSD, CCSD(T)]. Spectroscopic constants and dissociation energies are reported for the ground state of each species. The effects of relativity, electron correlation, and the basis set superposition error (BSSE) are investigated. At the BSSE corrected CCSD(T) level, the mean absolute error relative to experiment in D-e for three dimers (hydrides) is 0.13 (0.09) eV; for R-e the error is 0.024 (0.003) A, and for omega (e) it is 2 (13) cm-1. These illustrative calculations confirm that the present basis sets fulfill their design objectives.