I will present some generalizations of the one-dimensional forest fire model with ignition occurring only at zero, which was analyzed in particular by Volkov (2009). In more recent work, we allow the rates at which the trees grow to depend on their location, and the fire can spread using long-range connections. We establish that the expected time required for the fire to reach a site at a distance $x$ from the origin is at most of the order $l\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}g\phantom{\rule{0}{0ex}}x)l\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}g2+o(1)$. The talk is based on a recent joint paper with Mikhail Menshikov (Durham) and the late Francis Comets (Paris).