  We present the AM$^3$ (``Astrophysical Multi-Messenger Modeling'') software,
which has been successfully used in the past to simulate the multi-messenger
emission, including neutrinos, from active galactic nuclei, including the
blazar sub-class, gamma-ray bursts, and tidal disruption events. AM$^3$ is a
documented state-of-the-art open source software that efficiently solves the
coupled integro-differential equations for the spectral and temporal evolution
of the relevant particle densities (photons, electrons, positrons, protons,
neutrons, pions, muons, and neutrinos). AM$^3$ includes all relevant
non-thermal processes (synchrotron, inverse Compton scattering, photon-photon
annihilation, proton-proton and proton-photon pion production, and photo-pair
production). The software self-consistently calculates the full cascade of
primary and secondary particles, outperforming simple test-particle approaches,
and allows for non-linear feedback and predictions in the time domain. It also
allows to track separately the contributions of different radiative processes
to the overall photon and neutrino spectra, including the different hadronic
interaction channels. With its efficient hybrid solver combining analytical and
numerical techniques, AM$^3$ combines efficiency and accuracy at a
user-adjustable level. We describe the technical details of the numerical
framework and present examples of applications to various astrophysical
environments.
