While the celebrated singularity theorems of Penrose and Hawking ensure the existence of singularities of space-time solutions of the Einstein equations under rather general conditions, the precise structure of these singularities has only been partially investigated. Previously, the most exhaustive analysis was due to V. A. Belinskii, I. M. Khalatnikov, and E. M. Lifshitz, viz. BKL, who in various papers from 1970 to 1982, suggested that subject to certain asymptotic assumptions generic singularities are oscillatory and may exhibit very complicated behavior. The present paper offers a rigorous mathematical analysis of the heuristic BKL arguments, and establishes two general theorems for quiescent, i.e. non-oscillatory, singularities in velocity dominated systems when the Einstein equations are coupled to a scalar field, and a stiff fluid respectively. Their results show that a neighborhood of a singularity can be covered by a Gaussian coordinate system in which the singularity is simultaneous and its evolution decouples at different spatial points. This conclusion is consistent with the strong censorship hypothesis, and seems to generally confirm the BKL results. Contents include: an introduction (which gives a lucid overview); the main results (which state the two major theorems and describe their mathematical preliminaries); the framework of their proofs; Fuchsian systems; setting up of the reduced equations; the curvature estimates; the constraints; and finally a discussion.