  We introduce and apply a numerically exact method for investigating the
real-time dissipative dynamics of quantum impurities embedded in a macroscopic
environment beyond the weak-coupling limit. We focus on the spin-boson
Hamiltonian that describes a two-level system interacting with a bosonic bath
of harmonic oscillators. This model is archetypal for investigating dissipation
in quantum systems and tunable experimental realizations exist in mesoscopic
and cold-atom systems. It finds abundant applications in physics ranging from
the study of decoherence in quantum computing and quantum optics to extended
dynamical mean-field theory. Starting from the real-time Feynman-Vernon path
integral, we derive an exact stochastic Schr\"odinger equation that allows to
compute the full spin density matrix and spin-spin correlation functions beyond
weak coupling. We greatly extend our earlier work (P. P. Orth, A. Imambekov,
and K. Le Hur, Phys. Rev. A {\bf 82}, 032118 (2010)) by fleshing out the core
concepts of the method and by presenting a number of interesting applications.
Methodologically, we present an analogy between the dissipative dynamics of a
quantum spin and that of a classical spin in a random magnetic field. This
analogy is used to recover the well-known non-interacting-blip-approximation in
the weak-coupling limit. We explain in detail how to compute spin-spin
autocorrelation functions. As interesting applications of our method, we
explore the non-Markovian effects of the initial spin-bath preparation on the
dynamics of the coherence $\sigma^x(t)$ and of $\sigma^z(t)$ under a
Landau-Zener sweep of the bias field. We also compute to a high precision the
asymptotic long-time dynamics of $\sigma^z(t)$ without bias and demonstrate the
wide applicability of our approach by calculating the spin dynamics at non-zero
bias and different temperatures.
