Hey, MightyOwl! Catalina should be supported for the Blizzard Application now since it’s live! There are a few issues that we are aware of though.
- Diablo II will not run on Catalina at the moment, please see this forum thread. Our team is aware, but there is no ETA on how long this may take to be compatible with Catalina.
- Heroes of the Storm and StarCraft II are experiencing kernel panic crashes for those with Radeon Graphics on Mac OS on the Metal Graphics API. This is a driver issue which will hopefully be fixed in a future MacOS update. Our team is working on resolving these concerns.
Here’s the forum thread for those issues .
There’s also issues with CMD+TAB for Heroes of the Storm causing black screen and needing a reboot, windowed mode will work around this issue. Both games and other versions of MacOS will sometimes experience Quit Unexpectedly, which changing the Computer name will address. - ^ This may also be impacting graphical pixelation issues for Diablo III on OpenGL Graphics API. Here’s the thread for this issue.
All other games should be running fairly well though and compatible. If there are any issues, please create a new thread or contact our Technical supports to troubleshoot these issues individually 
Hopefully, this gets all the issues known at the moment. TLDR; Diablo II is not yet compatible. If the system has a Radeon Graphics card, you may run into issues with some of our games.