Overview
Worlds become unlockable once you gather a certain amount of gold in the arena. There is a separate arena guide on Steam that I used to unlock them, and recommend following that.
You must complete each world once before you can proceed to the next one on the list (going up the image below, vertically). Each world adds its own additive permanent bonuses to all worlds (so if you do water world twice, then the bonus is 100x + 100x = 200x drowner power, not 100x * 100x = 10,000x), and worlds beyond the basic also have some downsides that increase challenge as well.
Basic World
You can return to the basic world after you've gone to the other worlds. Because it has no difficulty multiplier, the reasons you might do that are as follows:
1. To complete challenges with the bonuses from future worlds
2. To farm arena multiplier
3. To farm undead secondaries
Water World
Fire World
Undead World
Dark World
Rusty World
Source: https://steamcommunity.com/sharedfiles/filedetails/?id=3359596557
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