Max -infinitely divisible (max -id) processes play a central role in extreme -value theory and include the subclass of all max -stable processes. They allow for a constructive representation based on the pointwise maximum of random functions drawn from a Poisson point process defined on a suitable function space. Simulating from a max -id process is often difficult due to its complex stochastic structure, while calculating its joint density in high dimensions is often numerically infeasible. Therefore, exact and efficient simulation techniques for max -id processes are useful tools for studying the characteristics of the process and for drawing statistical inferences. Inspired by the simulation algorithms for max -stable processes, theory and algorithms to generalize simulation approaches tailored for certain flexible (existing or new) classes of max -id processes are presented. Efficient simulation for a large class of models can be achieved by implementing an adaptive rejection sampling scheme to sidestep a numerical integration step in the algorithm. The results of a simulation study highlight that our simulation algorithm works as expected and is highly accurate and efficient, such that it clearly outperforms customary approximate sampling schemes. As a by-product, new max -id models, which can be represented as pointwise maxima of general location -scale mixtures and possess flexible tail dependence structures capturing a wide range of asymptotic dependence scenarios, are also developed. (c) 2022 EcoSta Econometrics and Statistics.