Android (rooted): LTE is pretty handy when you're browsing the web or downloading files, but when you aren't actively using your phone, it draws a lot of battery without being useful. Smart Network switches your phone to 2G when your screen is off to save battery while keeping you connected.
You'll need the Xposed framework installed for this to work, but it's totally worth it. Install it, head to the Download section, and search for Smart Network to install the tweak. You'll have to reboot your phone, but you can then access it from the app drawer to adjust its settings.
Smart Network gives you the option to set your default network mode when the screen is on and off, which may require a bit of research to see what your phone supports. You can also set it to disconnect or turn off Wi-Fi when your screen is off if you so desire. You can even whitelist certain apps, so your network won't switch to 2G if a certain app is doing something important (like streaming music).
I was hesitant to try this out, assuming that switching between 2G and LTE would take too long for it to be practical -- but it's actually quite seamless. When I unlock my phone, it takes only a second or two to switch to LTE -- fast enough that I don't even notice it, and battery life has been stellar so far. And, considering I only need basic connectivity when my screen is off, it doesn't feel like I've made any sacrifice -- all my emails and notifications still come in beautifully.
Popular battery-saving app JuiceDefender has had this feature for a while, but only in the $4.99 Ultimate version. JuiceDefender contains a bunch of other features as well, but if all you want is this one tweak, Smart Network is great free alternative, especially if you're already using Xposed.
Read more about the app below, or download the Xposed framework now to try it out.