In this talk, we will describe work of the speaker along with Leise Borg, Juergen Frikel, and Jakob Jorgensen characterizing how artifacts appear in limited data X-ray tomography with arbitrary data sets. We also provide estimates of the strength of the added artifacts in some cases. We will illustrate our theoretical predictions using reconstructions from real and simulated data for standard and nonstandard limited CT problems. We will put our results in a general mathematical framework that can be used on a range of limited data problems, and we will mention the key ideas in the proof, which is based on microlocal analysis. This work is motivated by an unusual synchrotron CT data set that we will show that has artifacts not seen in standard limited data problems.