Work your way from the four single-elementals to an Island full of all the Monsters on Air Island. Click the Breeding Structure in the top-left, then two Monsters (besides Epics or other Monsters that can't breed normally) and the result will (hopefully) be a new Monster depending on which two Monsters you bred. Once the breeding has finished, you can click the Breeding Structure again to send the Monster to the Nursery, where it will hatch. After the Monster has finished hatching, click the Nursery to place the Monster down if you haven't already placed that Monster down. Occasionally you'll get Rares instead of Commons. Epic Monsters have unique combinations and cannot be selected to breed. Each Monster has its own sound (Rares and Epics have their own sounds except for Rare/Epic Wubbox). Collect 100 duplicate Monsters to obtain the Dipsters. Collect 150 duplicate Monsters to enhance your Breeding Structure & Nursery, making eggs be bred & hatch faster. Collect 200 duplicate Monsters to obtain the Werdos (Parlsona & Stoowarb).
