  For a graph $H$, the Tur\'{a}n number of $H$, denoted by ex$(n,H)$, is the
maximum number of edges of an $n$-vertex $H$-free graph. Let $g(n,H)$ denote
the maximum number of edges not contained in any monochromatic copy of $H$ in a
$2$-edge-coloring of $K_n$. A wheel $W_m$ is a graph formed by connecting a
single vertex to all vertices of a cycle of length $m-1$. The Tur\'{a}n number
of $W_{2k}$ was determined by Simonovits in the 1960s. In this paper, we
determine ex$(n,W_{2k+1})$ when $n$ is sufficiently large. We also show that,
for sufficiently large $n$, $g(n,W_{2k+1})=\mbox{ex}(n,W_{2k+1})$ which
confirms a conjecture posed by Keevash and Sudakov for odd wheels.
