  Monte Carlo simulations of lattice quantum field theories on Lefschetz
thimbles are non trivial. We discuss a new Monte Carlo algorithm based on the
idea of computing contributions to the functional integral which come from
complete flow lines. The latter are the steepest ascent paths attached to
critical points, i.e. the basic building blocks of thimbles. The measure to
sample is thus dictated by the contribution of complete flow lines to the
partition function. The algorithm is based on a heat bath sampling of the
gaussian approximation of the thimble: this defines the proposals for a
Metropolis-like accept/reject step. The effectiveness of the algorithm has been
tested on a few models, e.g. the chiral random matrix model. We also discuss
thimble regularization of gauge theories, and in particular the successfull
application to 0+1 dimensional QCD and the status and prospects for Yang-Mills
theories.
