  In this paper, we study the existence of distributional solutions of the
following non-local elliptic problem \begin{eqnarray*}
  \left\lbrace
  \begin{array}{l}
  (-\Delta)^{s}u + |\nabla u|^{p} =f \quad\text{ in } \Omega
  \qquad \qquad \qquad \,\,\, u=0 \,\,\,\,\,\,\,\text{ in }
\mathbb{R}^{N}\setminus \Omega, \quad s \in (1/2, 1).
  \end{array}
  \right. \end{eqnarray*}
  We are interested in the relation between the regularity of the source term
$f$, and the regularity of the corresponding solution. If $p<2s$, that is the
natural growth, we are able to show the existence for all $f\in L^1(\O)$. In
the subcritical case, that is, for $p < p_{*}:=N/(N-2s+1)$, we show that
solutions are $\mathcal{C}^{1, \alpha}$ for $f \in L^{m}$, with $m$ large
enough. In the general case, we achieve the same result under a condition on
the size of the source. As an application, we may show that for regular
sources, distributional solutions are viscosity solutions, and conversely.
