Lightning Returns: Final Fantasy XIII is a game of failure. It stands to reason, too, a the world is about to end. Who wouldn't expect a little bit of fail? Consequently, it's expected that you will have to play through the game at least twice before you've seen everything, because on your first try you just won't be capable of getting everything done.
That's where New Game + comes in. Bless you, Chrono Trigger, for inventing New Game +.
Essentially a do-over of the entire game with the stats and items of your previous attempt, a New Game + will allow you to go back and beat those parts of Lightning Returns: Final Fantasy XIII that trounced you before. This article will explore the ways in which New Game + is implemented in Lightning Returns, and how to use them to your advantage.
There are three ways in which you can begin a New Game +:
- Failing to complete the five main quests before the end of the world. If you reach the final day without having completed these quests, armageddon will cut you off from the final dungeon in the game, and you'll have to start over from the first day.
- Using the Crystal of Atonement. The Crystal of Atonement appears beside the door to the final boss if you fail to beat him on your first try. You can use it to scrap your current game and begin anew. The Crystal of Atonement will also appear in the Ark if you achieve the third method of triggering a New Game +...
- ... which is, of course, to beat the game.
There are a few differences in your New Game + depending on how it was triggered.
If you failed your quest or simply gave up, your New Game + will set you back to the first day. Lightning will retain her current stat boosts, equipment, items, gil, and schemata configurations. All quests will be reset, all key items will be taken away (this includes the Ultima Weapon and Ultima Shield if you managed to collect them in the final dungeon), and any progress towards reducing the numbers of a monster population to spawn a Last One will be lost. 
Any bosses fought in the previous game may still be battled in the Ark at any time. You may also change the game's difficulty level or keep it the same as before. Quests completed previously will still offer stat boosts, though they'll be lower than before, and new items will not be added to your inventory. Your EP cap continues to rise with completed quests to a maximum of nine; item slots continue to rise to a maximum of twelve.
In addition to the above, treasure spheres will all be reset - though with a twist. There's no point in having two of the same item, so any accessories you acquire in a New Game + will instead be upgraded to a new, more powerful form. The first time this happens they'll gain a '+' form; the second time this happens the name will change slightly. Try to collect an item a third time and (typically) you'll receive gil instead. The treasure boxes in the Ark, by contrast, will contain only gil if they've already been opened in a previous run.
Note that you will be transported directly to the Ark in your New Game +; the introductory battles in the Patron's Palace are skipped entirely.
If you completed the game, all of the above will still apply. Lightning keeps her items, quests are reset, monsters are reset, yadda yadda. As a further reward for beating the game, however, you gain a few extra tidbits which will make your next run through Lightning Returns a bit more interesting.
- You may now upgrade your weapons and shields at Forges. Weapons and shields can become significantly stronger as a result, and will allow Lightning to whip through formerly-tough enemies rather easily.
- You may now purchase upgrade items at Adventuring Essentials shops. This includes items used for leveling up magic and abilities (namely Malistones).
- You may now select Hard Mode when starting a new game. Hard Mode is, well, you know.
- Assuming you open the Ark's treasure box on the first day, Lightning will gain Proof of Overcoming Limits. This handy-dandy item, when activated, allows Lightning to break the usual damage limit of 99,999 HP on a single hit, ultimately allowing her to deliver 999,999 HP of damage. Kind of overkill for most enemies, but there you go. While Proof of Overcoming Limits is active you cannot register online records for killing bosses.
- The treasure boxes in the Ark will now typically contain upgrade items, such as Malistones or Mighty Material, rather than gil.
- Completed quests will offer more gil than normal.
The nice thing about New Game + is, largely, variety. With her increased strength Lightning can now take on quests in just about any order she chooses, allowing you to engage in a much more free-form experience over focusing your efforts on becoming strong enough to kill bosses. Explore, experiment, and dominate the heck out of Nova Crysalia.