Epic Bosses are Villains that will only appear once all threats (Low, Medium, High Threats, and Bosses) and Deploys in a Mission have been completed. Some of the Deploys require a hero of a specific Class or a specific Hero to be completed. Epic Bosses appear in Missions with the purple crown icon.
If any of the Deploys have not been completed when the last battle is done, the Epic Boss will not appear. To prevent this, players should do the Deploy Mission as soon as it appears.