  We develop a theoretical approach to ``spontaneous stochasticity'' in
classical dynamical systems that are nearly singular and weakly perturbed by
noise. This phenomenon is associated to a breakdown in uniqueness of solutions
for fixed initial data and underlies many fundamental effects of turbulence
(unpredictability, anomalous dissipation, enhanced mixing). Based upon analogy
with statistical-mechanical critical points at zero temperature, we elaborate a
renormalization group (RG) theory that determines the universal statistics
obtained for sufficiently long times after the precise initial data are
``forgotten''. We apply our RG method to solve exactly the ``minimal model'' of
spontaneous stochasticity given by a 1D singular ODE. Generalizing prior
results for the infinite-Reynolds limit of our model, we obtain the RG fixed
points that characterize the spontaneous statistics in the near-singular,
weak-noise limit, determine the exact domain of attraction of each fixed point,
and derive the universal approach to the fixed points as a singular
large-deviations scaling, distinct from that obtained by the standard
saddle-point approximation to stochastic path-integrals in the zero-noise
limit. We present also numerical simulation results that verify our analytical
predictions, propose possible experimental realizations of the ``minimal
model'', and discuss more generally current empirical evidence for ubiquitous
spontaneous stochasticity in Nature. Our RG method can be applied to more
complex, realistic systems and some future applications are briefly outlined.
