We study the multi-view geometry of 1D radial cameras. A broad a class of both central and non-central cameras, such as fish-eye and catadioptric cameras, can be reduced to 1D radial cameras under the assumption of known center of radial distortion. For cameras in general configuration, we introduce a quadrifocal tensor that can be computed linearly from 15 or more features seen in four views. From this tensor a metric reconstruction of the 1D cameras as well as the observed features can be obtained. In a second phase this reconstruction can then be used as a calibration object to estimate a non-parametric non-central model for the cameras. We study some degenerate cases, including pure rotation. In the case of a purely rotating camera we obtain a trifocal tensor that can be estimated linearly from 7 points in three views. This allows us to obtain a metric reconstruction of the plane at infinity. Next, we use the plane at infinity as a calibration device to non-parametrically estimate the radial distortion. We demonstrate the results of our approach on real and synthetic images.