  In many interacting particle systems, relaxation to equilibrium is thought to
occur via the growth of 'droplets', and it is a question of fundamental
importance to determine the critical length at which such droplets appear. In
this paper we construct a mechanism for the growth of droplets in an arbitrary
finite-range monotone cellular automaton on a $d$-dimensional lattice. Our main
application is an upper bound on the critical probability for percolation that
is sharp up to a constant factor in the exponent. Our method also provides
several crucial tools that we expect to have applications to other interacting
particle systems, such as kinetically constrained spin models on
$\mathbb{Z}^d$.
  This is one of three papers that together confirm the Universality Conjecture
of Bollob\'as, Duminil-Copin, Morris and Smith.
