Summary: In \( r\)-neighbour bootstrap percolation on a graph \( G\), a (typically random) set \( A\) of initially ‘infected’ vertices spreads by infecting (at each time step) vertices with at least \( r\) already-infected neighbours. This process may be viewed as a monotone version of the Glauber dynamics of the Ising model, and has been extensively studied on the \( d\)-dimensional grid \( [n]^d\). The elements of the set \( A\) are usually chosen independently, with some density \( p\), and the main question is to determine \( p_c([n]^d,r)\), the density at which percolation (infection of the entire vertex set) becomes likely. In this paper we prove, for every pair \( d,r \in \mathbb{N}\) with \( d \geq r \geq 2\), that \[ p_c( [n]^d,r ) = \left ( \frac {\lambda (d,r) + o(1)}{\log _{(r-1)} (n)} \right )^{d-r+1} \] as \( n \to \infty \), for some constant \( \lambda (d,r) > 0\), and thus prove the existence of a sharp threshold for percolation in any (fixed) number of dimensions. We moreover determine \( \lambda (d,r)\) for every \( d \geq r \geq 2\).