  We produce a list of 64 families of Fano fourfolds of K3 type, extracted from
our database of at least 634 Fano fourfolds constructed as zero loci of general
global sections of completely reducible homogeneous vector bundles on products
of flag manifolds. We study the geometry of these Fano fourfolds in some
detail, and we find the origin of their K3 structure by relating most of them
either to cubic fourfolds, Gushel-Mukai fourfolds, or actual K3 surfaces. Their
main invariants and some information on their rationality and on possible
semiorthogonal decompositions for their derived categories are provided.
