Microsoft (and Sony for that matter) may say that they are supporting the current gen systems for multiple years after the next gen is released but let's think practical here. Does "supporting" mean keeping all the servers open or making developers make games for it. Good games not just some code on a disk. I doubt that. And honestly I don't want to see a lot of current gen ports of the next gen games. I really feel like bungie could have made Destiny even greater if it would have only been a next gen title. I don't want my next gen games being handicapped because they have to make current(old) gen versions. I think both companies will still produce content for the current gen systems but their main focus will undoubtedly be on next gen. That being said I'm sure Destiny will still be an awesome game on xbox 360 so you shouldn't have anything to worry about. But if you want a next gen system with less of a next gen price then go with the ps4
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Considering they have released another edition of the xbox 360 alongside of the announcement of Xbox One so yes, the console will be supported at least 2 to 3 years more. What Sony has planned, we dont know cause I dont pay attention to news about that system since I am not getting it. But Microsoft has no intention of stopping support, multiplayer support or any of that matter for the current Xbox 360 just because the Xbox One is being released.