I am using CyanogenMod 10 on my device and saw that CM10.1 (Android 4.2) is available on get.cm.
If I upgrade, do I have to wipe my phone or will it just work?
Or what problems will I have to expect if I don't factory reset it before?
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It just works, no need to wipe.
You don't need to wipe, according to a blog post on CyanogenMod's Google Plus channel:
Upgrading from CM10 to CM10.1
As you may have noticed, we started updating some devices from cm10 (Android 4.1) to 10.1 (Android 4.2) during the weekend. The upgrade process has been tested extensively for the older devices (manta and mako didn't have that problem since they started out with 10.1), so you should be able to update directly without losing any data.
The single application known not to work is the Clock: Our changes to 4.1's clock to add the Stopwatch and Timer in CM10 aren't compatible with Google's own implementation of those features in 4.2 (which we're using in cm10.1); If your clock starts crashing after upgrading to 10.1, simply go into the Settings -> Apps menu, choose Clock from the All tab, and hit "Clear data".