Various approaches to products of residue currents

We describe various approaches to Coleff-Herrera products of residue currents $R^j$ (of Cauchy-Fantappi\`e-Leray type) associated to holomorphic mappings $f_j$. More precisely, we study to which extent (exterior) products of natural regularizations of the individual currents $R^j$ yield regularizations of the corresponding Coleff-Herrera products. Our results hold globally on an arbitrary pure-dimensional complex space.