  The burning process on a graph $G$ starts with a single burnt vertex, and at
each subsequent step, burns the neighbors of the currently burnt vertices, as
well as one other unburnt vertex. The burning number of $G$ is the smallest
number of steps required to burn all the vertices of the graph. In this paper,
we examine the problem of computing the burning number in a geometric setting.
The input is a set of points $P$ in the Euclidean plane. The burning process
starts with a single burnt point, and at each subsequent step, burns all the
points that are within a distance of one unit from the currently burnt points
and one other unburnt point. The burning number of $P$ is the smallest number
of steps required to burn all the points of $P$. We call this variant
\emph{point burning}. We consider another variant called \emph{anywhere
burning}, where we are allowed to burn any point of the plane. We show that
point burning and anywhere burning problems are both NP-complete, but
$(2+\varepsilon)$ approximable for every $\varepsilon>0$. Moreover, if we put a
restriction on the number of burning sources that can be used, then the
anywhere burning problem becomes NP-hard to approximate within a factor of
$\frac{2}{\sqrt{3}}-\varepsilon$.
