How to install Cyanogen without losing data I've been using ICS on my Samsung Galaxy S3 for a while now, but I can't update to Jelly Bean for multiple reasons. 
I've decided to switch to Cyanogen Mod, is there any way I can move to Cyanogen without losing any application data?
As geff described, you cannot avoid a wipe/factory-reset. But you can do a proper backup beforehand, so you can restore most of your data afterwards. This should work without any trouble for the apps you've installed yourself ("user apps"), while special care must be taken for pre-installed apps ("system apps"): As data structures (inside their databases) might have changed, a simple restore (replacing the databases) might break things and even render those apps unusable.

Provided your device is rooted, Titanium Backup is a good choice. Backup all "user apps" and their data, plus the "system apps" and their data. Special care on restore: "user apps" can be restored 1:1. Before goint to the "system apps", make sure to turn on TB's "migration" mode. Then restore only data, and do so one-by-one, after each "data set" verifying there are no side-effects.

An alternative to be named is Helium Backup, which even works on non-rooted devices.

Safest is to stick to your "user apps", and let contacts etc. solve via sync. For several things like SMS, there are special backup/restore solutions which you could use.