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Update September 2, 2014: Canonical released Ubuntu Developer Tools Center which can be used to easily install Android Studio and Android SDK in Ubuntu. That's the official, supported method of installing Android Studio in Ubuntu now!

Google provides Android Studio for Linux as a simple archive which you must download, extract, create your own launcher and so on.

To makes things easier, +Paolo Rotolo has created an Android Studio PPA for Ubuntu and derivatives (Linux Mint, elementary OS and so on).

The PPA is just for installing Android Studio, not for updating it, because Android Studio has its own update "manager". To update Android Studio to the latest version, simply click "Check for updates now" in the welcome screen, then click "Update and Restart":

It's important to note that the package tries to install one of the following JDKs: default-jdk, oracle-java7-installer or oracle-java8-installer. I saw in some comments that Android Studio users receive the following error: "OpenJDK 6 is not supported. Please use Oracle Java or newer OpenJDK", even after installing OpenJDK7. That's why I recommend using Android Studio with Oracle Java 7 or Oracle Java 8 from our PPA (see links at the bottom of the article).

And one more thing: Android Studio uses a low resolution icon and that can't be fixed (it would be overwritten when the app is updated through its own update mechanism anyway).

sudo add-apt-repository ppa:paolorotolo/android-studio
sudo apt-get update
sudo apt-get install android-studio