  We study a new geometric bootstrap percolation model, line percolation, on
the $d$-dimensional integer grid $[n]^d$. In line percolation with infection
parameter $r$, infection spreads from a subset $A\subset [n]^d$ of initially
infected lattice points as follows: if there exists an axis-parallel line $L$
with $r$ or more infected lattice points on it, then every lattice point of
$[n]^d$ on $L$ gets infected, and we repeat this until the infection can no
longer spread. The elements of the set $A$ are usually chosen independently,
with some density $p$, and the main question is to determine $p_c(n,r,d)$, the
density at which percolation (infection of the entire grid) becomes likely. In
this paper, we determine $p_c(n,r,2)$ up to a multiplicative factor of $1+o(1)$
and $p_c(n,r,3)$ up to a multiplicative constant as $n\rightarrow \infty$ for
every fixed $r\in \mathbb{N}$. We also determine the size of the minimal
percolating sets in all dimensions and for all values of the infection
parameter.
