In recent years\, a great deal of progress ha s been made in understanding the behaviour of a p articular class of monotone cellular automata\, co mmonly known as bootstrap percolation. In particu lar\, if one considers only two-dimensional autom ata\, then we now have a fairly precise understand ing of the typical evolution of these processes\, starting from p-random initial conditions of inf ected sites. Given a bootstrap model\, one can con sider the associated kinetically constrained spin model in which the state (infected or healthy) of a vertex is resampled (independently) at rate 1 by tossing a p-coin if it could be infected in th e next step by the bootstrap process\, and remains in its current state otherwise. Here p is the pr obability of infection. The main interest in KCM& rsquo\;s stems from the fact that\, as p &rarr\; 0 \, they mimic some of the most striking features of the glass transition\, a major and still largel y open problem in condensed matter physics. In th is talk\, motivated by recent universality results for bootstrap percolation\, I&rsquo\;ll discuss some &ldquo\;universality conjectures&rdquo\; con cerning the growth of the (random) infection time of the origin in a KCM as p &rarr\; 0. Joint proj ect with R. Morris (IMPA) and C. Toninelli (Paris VII)\, LOCATION:Seminar Room 1\, Newton Institute CONTACT:INI IT END:VEVENT END:VCALENDAR