  In this paper we study the following class of fractional Choquard--type
equations \[ (-\Delta)^{1/2}u + u=\Big( I_\mu \ast F(u)\Big)f(u), \quad
x\in\mathbb{R}, \] where $(-\Delta)^{1/2}$ denotes the $1/2$--Laplacian
operator, $I_{\mu}$ is the Riesz potential with $0<\mu<1$ and $F$ is the
primitive function of $f$. We use Variational Methods and minimax estimates to
study the existence of solutions when $f$ has critical exponential growth in
the sense of Trudinger--Moser inequality.
