decompiling DEX into Java sourcecode

Question

How can one decompile Android DEX (VM bytecode) files into corresponding Java sourcecode?

Answer 1

^{A more complete version of fred's answer:}

Manual way

First you need a tool to extract all the (compiled) classes on the DEX to a JAR.
There's one called dex2jar, which is made by a chinese student.

Then, you can use jd-gui to decompile the classes on the JAR to source code.
The resulting source should be quite readable, as dex2jar applies some optimizations.

Automatic way

You can use APKTool. It will automatically extract all the classes (.dex), resources (.asrc), then it will convert binary XML to human-readable XML, and it will also dissassemble the classes for you.
Disassembly will always be more robust than decompiling, especially with
JARs obfuscated with Pro Guard!

Just tell APKTool to decode the APK into a directory, then modify what you want,
and finally encode it back to an APK. That's all.

Important: APKTool dissassembles. It doesn't decompile.
The generated code won't be Java source.
But you should be able to read it, and even edit it if you're familiar with jasmin.
If you want Java source, please go over the Manual way.