CERN-LHC. This paper presents a study of $WW\gamma$ and $WZ\gamma$ triboson production using events from proton--proton collisions at a centre-of-mass energy of $\sqrt{s} = 8$ TeV recorded with the ATLAS detector at the LHC and corresponding to an integrated luminosity of 20.2 fb$^{-1}$. The $WW\gamma$ production cross-section is determined using a final state containing an electron, a muon, a photon, and neutrinos ($e\nu\mu\nu\gamma$). Upper limits on the production cross-section of the $e\nu\mu\nu\gamma$ final state and the $WW\gamma$ and $WZ\gamma$ final states containing an electron or a muon, two jets, a photon, and a neutrino ($e\nu jj\gamma$ or $\mu\nu jj\gamma$) are also derived. The results are compared to the cross-sections predicted by the Standard Model at next-to-leading order in the strong-coupling constant. In addition, upper limits on the production cross-sections are derived in a fiducial region optimised for a search for new physics beyond the Standard Model. The results are interpreted in the context of anomalous quartic gauge couplings using an effective field theory. Confidence intervals at 95% confidence level are derived for the 14 coupling coefficients to which $WW\gamma$ and $WZ\gamma$ production are sensitive. The $e\nu\mu\nu\gamma$ fiducial phase space region is defined as (Table 3 of the preprint): - 1 electron and 1 muon with opposite charge and $p_{\text{T}}^{\ell} > \mbox{20 GeV} $ each - no $3^{\text{rd}}$ lepton ($p_{\text{T}}^{\ell} > \mbox{7 GeV}$) - $\eta^{\ell} < 2.5$ - $\Delta R (\ell, \ell^{\prime}) < 0.1$ - at least one photon with $E_{\text{T}}^{\gamma} > \mbox{15 GeV}$ and $|\eta^{\gamma}| < 2.37$ - isolation fraction $\epsilon_{h}^{p} < 0.5$ - $\Delta R(\ell, \gamma) > 0.5$ - $E_{\text{T,rel}}^{\text{miss}} > \mbox{15 GeV}$ - $m_{e\mu}> \mbox{50 GeV}$ - $N_{\text{jets}} = 0$ with $p_{\text{T}}^{j} > \mbox{25 GeV}$ and $|y^{j}| < 4.4$ - $\Delta R(\text{jet}, \gamma) > 0.5$ - $\Delta R(\text{jet}, \ell) > 0.3$ The $\ell\nu jj\gamma$ fiducial phase space region with $\ell = e \text{ or }\mu$ is defined as (Table 3 of the preprint): - 1 electron or 1 muon with $p_{\text{T}}^{\ell} > \mbox{25 GeV}$ and $|\eta^{\ell}| < 2.5$ - no $2^{\text{nd}}$ lepton ($p_{\text{T}}^{\ell} > \mbox{7 GeV}$) - at least one photon with $E_{\text{T}}^{\gamma} > \mbox{15 GeV}$ and $|\eta^{\gamma}| < 2.37$ - isolation fraction $\epsilon_{h}^{p} < 0.5$ - $\Delta R(\ell, \gamma) > 0.5$ - $E_{\text{T}}^{\text{miss}} > \mbox{30 GeV}$ - $m_{\text{T}} > \mbox{30 GeV}$ - at least two jets with $p_{\text{T}}^{j} > \mbox{25 GeV}$ and $|\eta^{j}| < 2.5 $ - $N_{\text{$b$-jets}} = 0$ - $|\Delta \eta_{jj}| < 1.2$ - $\Delta R_{jj} < 3.0$ - $\mbox{70 GeV} < m_{jj} < \mbox{100 GeV}$ - $\Delta R(\text{jet}, \gamma) > 0.5$ - $\Delta R(\text{jet}, \ell) > 0.3$