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We studied the magnetic properties and electronic structure of $CeAuAl_3$ using density functional theory. This compound shows a large Sommerfeld coefficient, a Kondo temperature, ${T_K=4\,K}$ and antiferromagnetic order below ${T_N=1.1\,K}$. We calculated the magnetic groundstate of ${CeAuAl_3}$ and the magnetic anisotropy energies. Treating the 4f-electrons as localized with DFT+U we obtain a good match with the magnetic properties observed experimentally. We also report salient features of the electronic structure of ${CeAuAl_3}$, including features of the Fermi surface and associated quantum oscillatory spectra, when the 4f-electrons are treated either as localized or itinerant.