  Given two graphs $H_1$ and $H_2$, a graph is $(H_1,\,H_2)$-free if it
contains no induced subgraph isomorphic to $H_1$ or $H_2$. For a positive
integer $t$, $P_t$ is the chordless path on $t$ vertices. A paraglider is the
graph that consists of a chorless cycle $C_4$ plus a vertex adjacent to three
vertices of the $C_4$. In this paper, we study the structure of ($P_5$,
paraglider)-free graphs, and show that every such graph $G$ satisfies
$\chi(G)\le \lceil \frac{3}{2}\omega(G) \rceil$, where $\chi(G)$ and
$\omega(G)$ are the chromatic number and clique number of $G$, respectively.
Our bound is attained by the complement of the Clebsch graph on 16 vertices.
More strongly, we completely characterize all the ($P_5$, paraglider)-free
graphs $G$ that satisfies $\chi(G)> \frac{3}{2}\omega(G)$. We also construct an
infinite family of ($P_5$, paraglider)-free graphs such that every graph $G$ in
the family has $\chi(G)=\lceil \frac{3}{2}\omega(G) \rceil-1$. This shows that
our upper bound is optimal up to an additive constant and that there is no
$(\frac{3}{2}-\epsilon)$-approximation algorithm to the chromatic number of
($P_5$, paraglider)-free graphs for any $\epsilon>0$.
