Abstract: The extragalactic gamma-ray sky is dominated by blazars, a subclass of active galaxies with jets extending to millions of light years in distance that are closely aligned to our line of sight. Blazar jets are magnetized outflows of plasma, traveling almost at the speed of light and “shining” across the electromagnetic spectrum. Their broadband radiation is variable on timescales ranging from several months to a few minutes! It is established that the low-energy radiation (radio up to X-rays) is produced by relativistic leptons (electrons and positrons) gyrating in the jet’s magnetic fields, but the origin of gamma-ray emission remains a mystery. Blazars are one of the few astrophysical sources that can accelerate and confine the most energetic hadrons (i.e., protons and heavy ions) that we detect on Earth as high-energy cosmic rays. The smoking gun of hadronic acceleration in jets are neutrinos – neutral and nearly massless particles produced by interactions of relativistic hadrons with matter and radiation. In 2017, the IceCube Neutrino Observatory reported the detection of a neutrino in coincidence with a gamma-ray flare from a blazar at a distance of 5.7 billion light years from Earth. The time is therefore ripe to take stock of the current status in multi-messenger astrophysics and address some long-standing questions: Can we identify the jet conditions causing the variable electromagnetic emission? Are there “leptonic” and “hadronic” blazars? Are neutrinos produced preferentially during gamma- ray flares? Using state-of-the-art numerical codes and data analysis techniques we will make theoretical predictions for time-dependent multi-messenger emissions of blazars. Particularly, we will perform a comprehensive temporal study of electromagnetic and neutrino emissions in leptonic and hadronic scenarios, create a large suite of simulated gamma-ray light curves and spectra for the next-generation Cherenkov Telescope Array, and develop a novel hadronic fitting tool for blazar emission.