We start this series of lectures on quantum Monte Carlo methods with the simplest model for a strongly correlated system, namely an antiferromagnetic HeisenbergS‐1/2 chain. We will review methods for its simulation starting with the world‐line algorithm and then introducing the loop‐algorithm with global updates. We discuss next a model for doped antiferromagnets, with emphasis on the strong correlation limit that is central for high temperature superconductors and related materials, namely the so‐called t‐J model. An exact canonical transformation for this model that leads to a formulation with separated charge and spin degrees of freedom will be discussed. Results for a single hole in antiferromagnetic chains and planes will be summarized, giving an initial picture of charge dynamics in a quantum antiferromagnet. An additional algorithmic element, namely the determinantal method, will be introduced next, in order to deal with a finite number of charge degrees of freedom. Merging the loop‐ and the determinantal methods leads to a new one, the hybrid‐loop algorithm, that combines the efficiency of the loop‐algorithm for spins with the one of the determinantal method for fermions. Finally we discuss a first application of the hybrid‐loop algorithm, where the spectral function of the t‐J model in one dimension with finite doping is determined. There we can see how charge spin separation shows up in the one‐particle spectral function, and a detailed description can be achieved by comparison with an analitically soluble model, namely the t‐J model with 1/r2interaction. The results show that in addition to spinons and holons expected in one‐dimensional metals, antiholons with chargeQ= 2e, spinS= 0, and twice the mass of the holons are necessary to describe the inverse photoemission spectra at finite doping. © 2004 American Institute of Physics