The authors study spins with correlated ('bootstrap') dilution in which a (randomly occupied) site only develops a magnetic moment if surrounded by at least m magnetic neighbours. A generalisation of the Harris criterion indicates that the tendency of correlated disorder is to favour a new critical behaviour upon dilution. They also present real-space renormalisation group calculations for phase diagrams and discuss the different approximations involved. The structure of the infinite cluster at the percolation threshold of bootstrap dilution is compared with those of fractal geometries. Other correlated percolation models are compared with the present model.