  The burning number of a graph $G$ is the smallest number $b$ such that the
vertices of $G$ can be covered by balls of radii $0, 1, \dots, b-1$. As
computing the burning number of a graph is known to be NP-hard, even on trees,
it is natural to consider polynomial time approximation algorithms for the
quantity. The best known approximation factor in the literature is $3$ for
general graphs and $2$ for trees. In this note we give a
$2/(1-e^{-2})+\varepsilon=2.313\dots$-approximation algorithm for the burning
number of general graphs, and a PTAS for the burning number of trees and
forests. Moreover, we show that computing a
$(\frac53-\varepsilon)$-approximation of the burning number of a general graph
$G$ is NP-hard.
