  In this paper we study in complete generality the family of two-state,
deterministic, monotone, local, homogeneous cellular automata in $\mathbb{Z}^d$
with random initial configurations. Formally, we are given a set
$\mathcal{U}=\{X_1,\dots,X_m\}$ of finite subsets of
$\mathbb{Z}^d\setminus\{\mathbf{0}\}$, and an initial set
$A_0\subset\mathbb{Z}^d$ of `infected' sites, which we take to be random
according to the product measure with density $p$. At time $t\in\mathbb{N}$,
the set of infected sites $A_t$ is the union of $A_{t-1}$ and the set of all
$x\in\mathbb{Z}^d$ such that $x+X\in A_{t-1}$ for some $X\in\mathcal{U}$. Our
model may alternatively be thought of as bootstrap percolation on
$\mathbb{Z}^d$ with arbitrary update rules, and for this reason we call it
$\mathcal{U}$-bootstrap percolation.
  In two dimensions, we give a classification of $\mathcal{U}$-bootstrap
percolation models into three classes -- supercritical, critical and
subcritical -- and we prove results about the phase transitions of all models
belonging to the first two of these classes. More precisely, we show that the
critical probability for percolation on $(\mathbb{Z}/n\mathbb{Z})^2$ is $(\log
n)^{-\Theta(1)}$ for all models in the critical class, and that it is
$n^{-\Theta(1)}$ for all models in the supercritical class.
  The results in this paper are the first of any kind on bootstrap percolation
considered in this level of generality, and in particular they are the first
that make no assumptions of symmetry. It is the hope of the authors that this
work will initiate a new, unified theory of bootstrap percolation on
$\mathbb{Z}^d$.
