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We study supersymmetric solutions in four-dimensional $N=6$ gauged supergravity with $SO(6)$ gauge group. There is a unique $N=6$ supersymmetric $AdS_4$ vacuum with $SO(6)$ symmetry dual to an $N=6$ SCFT in three dimensions. We find a number of domain walls interpolating between this $AdS_4$ vacuum and singular geometries in the IR with $SO(2)\times SO(4)$, $U(3)$, $SO(3)$ and $SO(2)\times SO(2)\times SO(2)$ symmetries. The $SO(3)$ case admits $N=6$ or $N=2$ solutions depending on whether the pseudoscalars are present or not. On the other hand, all the remaining solutions preserve $N=6$ supersymmetry. These solutions describe RG flows from the $N=6$ SCFT to non-conformal field theories driven by mass deformations. In particular, the $SO(2)\times SO(4)$ solution is in agreement with the previously known mass deformations of the dual $N=6$ SCFT. We also give a supersymmetric Janus solution with $SO(2)\times SO(4)$ symmetry, describing two-dimensional conformal defects in the $N=6$ SCFT with unbroken $N=(4,2)$ supersymmetry. Finally, we find an $N=2$ supersymmetric $AdS_2\times H^2$ solution with $SO(2)\times SO(4)$ symmetry and the corresponding domain wall interpolating between this fixed point and the $AdS_4$ vacuum. The solution describes an $AdS_4$ black hole with a magnetic charge and is dual to a twisted compactification of the $N=6$ SCFT on a hyperbolic space $H^2$. We also give a domain wall interpolating between a locally supersymmetric $AdS_4$ and a curved domain wall with $SO(2)\times SO(2)\times SO(2)$ symmetry dual to an RG flow across dimensions from the $N=6$ SCFT to a supersymmetric quantum mechanics.