I'm currently playing through this game (bit late to it, I know) and am wondering if the game will allow me to go back to previous regions and find all the collectables (Riddler Trophies, etc) after I've completed the story - or if I need to do these as I go through the game?
Yes, you can freeplay after the end of the game in order to collect all the Riddler trophies and complete any challenges you may have missed.
This answer is applicable to the PS3 version and may be applicable to the other versions as well.
The game does not continue, it ends immediately after the last boss fight; however you can roam around as you please just before it begins.
Spoiler about how you know you're about to reach the no-turning-back point:
When there are about 20 goons clapping and cheering for you just before the visitor's center. Once you open the door to the visitor's center, you cannot go back.
If I recall correctly there's also an auto-save just before the no-turning-back point so after finishing the game you can load it (also via "continue") and play from there - but it takes place before the final fight, so it's not a continuation.