● AndroidLost

This is the very 1st application which I install on any phone that I get my hands on because loosing your Android smartphone isn't fun (but it rhymes).
Though with the Google Playstore Device Manager (https://play.google.com/store -> Cog icon -> Device Manager) you can remotely locate your smartphone, make it ring, set up a lock and erase it, AndroidLost is in a totally different league.
I can't list here all the fantastic features of AndroidLost but here are some, which IMHO makes it so great:
• Though like any other Google app, you can remotely install it, you can also configure this app, even after you've lost your phone.
• You can get a full status of your phone (location, battery level, WiFi, GPS on/off, what WIFI network it is connected to, etc....)
• You can make you phone flash the screen, take pictures, record conversations, lock it, wipe it, recover files (pictures and so) stored on the SD-Card.
• You can find out what SMS and phone calls have been made and received.
• You can set message to appear when your phone boots up and/or other type of messages, allowing you tell the person who may have found your phone, how to 
  contact you.
• Best of all you don't a need a computer to remotely control your phone, just about everything can be all done by sending specific SMS to your phone, which is especially handy to make phone ring/flash to locate it more precisely when you are near it but can't see it. Aside from having had to use it and recovered my phone with it, you'll find some of the testimonials here.

Root neededNo but if you do, it is much easier to convert it into a System app, thus preventing it be maliciously un-installed.
Tips: Once you've installed it, do make sure that you go in Settings ► Personal ► Security ► Device Administrators and allow it.