It's possible to join ABS to PVC, but it's only supposed to be done with a mechanical coupling, not glued. If you just came here for the answer, you can stop reading... but I think you want the whole story. Here goes.
Today, there are two commonly used plastics for drain, waste, and vent pipes inside of homes; ABS (black) and PVC (white, mostly). This is almost all that's ever used in new residential construction here in Minnesota, and I suspect for the rest of the country as well. For most systems, a plumber will use one material or the other, not but both. This makes all of the connections simple; the plumber uses a solvent cement (aka glue) made just for that material, things stick together, and everyone is happy...
Until then the "handy" homeowner comes along and messes it up.


For the most part, ABS and PVC are not supposed to be glued together. We'll turn to the 2015 Minnesota State Plumbing Code for some background on this topic.
ABS to PVC: What the MN plumbing code says
If we turn to chapter 7 of the MN state plumbing code, we can find a bunch of info on drains. Section 705.1.2 ABS Solvent Cement Joints (for ABS) says the following, among other stuff:
So that says you can glue ABS using something that meets ASTM D 2235. That's the standard for solvent cement for ABS.
Section 705.7.2 Solvent Cement Joints (for PVC) says the following, among other stuff:
So as long as you use the right purple primer, you can also glue PVC. You need to use a glue that meets ASTM D 2564, the standard for solvent cement for PVC. Of course.