  Trawl processes are continuous-time, stationary and infinitely divisible
processes which can describe a wide range of possible serial correlation
patterns in data. In this paper, we introduce new simulation algorithms for
trawl processes with monotonic trawl functions and establish their error bounds
and convergence properties. We extensively analyse the computational complexity
and practical implementation of these algorithms and discuss which one to use
depending on the type of L\'evy basis. We extend the above methodology to the
simulation of kernel-weighted, volatility modulated trawl processes and develop
a new simulation algorithm for ambit fields. Finally, we discuss how simulation
schemes previously described in the literature can be combined with our methods
for decreased computational cost.
