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We suggest a general framework for simulation of the triplet $(X_T,\bar X_T,\tau_T)$ (L\'evy process, its extremum, and hitting time of the extremum),and, separately, $X_T,\bar X_ T$ and pairs $(X_T,\bar X_ T)$, $(\bar X_T,\tau_T)$,  $(\bar X_ T-X_T,\tau_T)$, via characteristic functions and conditionalcharacteristic functions. The conformal deformations technique allows one toevaluate probability distributions, joint probability distributions andconditional probability distributions accurately and fast. For simulations inthe far tails of the distribution, we precalculate and store the values of the(conditional) characteristic functions on multi-grids on appropriate surfacesin $C^n$, and use these values to calculate the quantiles in the tails. Forsimulation in the central part of a distribution, we precalculate the values ofthe cumulative distribution at points of a non-uniform (multi-)grid, and useinterpolation to calculate quantiles.

Simulation of a L\'evy process, its extremum, and hitting time of the extremum via characteristic functions