While supported by an overwhelming body of empirical evidence, the classical notion of molecular structure is hard to reconcile with the probabilistic nature of the quantum theory of indistinguishable particles. This conflict between the most fundamental concept in chemistry and the thoroughly established physical theory of the atomic-scale world has given rise to debates over several decades. The reconciliation adopted in theoretical and computational chemistry is the Born-Oppenheimer approximation (BOA) which, in essence, imposes the molecular-structure concept upon quantum theory by explicitly treating the atomic nuclei as fixed classical point charges, eventually leading to the molecular-orbital theories that elegantly rationalize observed chemical phenomena. The BOA thus is highly successful and, importantly, its mathematical validity and numerical accuracy is very well established. The fundamental conflict persists, however.