  The non-Markovian nature of rough volatility processes makes Monte Carlo
methods challenging and it is in fact a major challenge to develop fast and
accurate simulation algorithms. We provide an efficient one for stochastic
Volterra processes, based on an extension of Donsker's approximation of
Brownian motion to the fractional Brownian case with arbitrary Hurst exponent
$H \in (0,1)$. Some of the most relevant consequences of this `rough Donsker
(rDonsker) Theorem' are functional weak convergence results in Skorokhod space
for discrete approximations of a large class of rough stochastic volatility
models. This justifies the validity of simple and easy-to-implement Monte-Carlo
methods, for which we provide detailed numerical recipes. We test these against
the current benchmark Hybrid scheme~\cite{BLP17} and find remarkable agreement
(for a large range of values of~$H$). This rDonsker Theorem further provides a
weak convergence proof for the Hybrid scheme itself, and allows to construct
binomial trees for rough volatility models, the first available scheme (in the
rough volatility context) for early exercise options such as American or
Bermudan options.
