Is there a way to get the source code from an APK file?

Simple way: use online tool https://www.decompiler.com/, upload apk and get source code.

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Procedure for decoding .apk files, step-by-step method:

Step 1:

1. Make a new folder and copy over the .apk file that you want to decode.

2. Now rename the extension of this .apk file to .zip (e.g. rename from filename.apk to filename.zip) and save it. Now you can access the classes.dex files, etc. At this stage you are able to see drawables but not xml and java files, so continue.

Step 2:

1. Now extract this .zip file in the same folder (or NEW FOLDER).

2. Download dex2jar (Don't download the code, click on the releases button that's on the right then download that) and extract it to the same folder (or NEW FOLDER).

3. Move the classes.dex file into the dex2jar folder.

4. Now open command prompt and change directory to that folder (or NEW FOLDER). Then write d2j-dex2jar classes.dex (for mac terminal or ubuntu write ./d2j-dex2jar.sh classes.dex) and press enter. You now have the classes.dex.dex2jar file in the same folder.

5. Download java decompiler, double click on jd-gui, click on open file, and open classes.dex.dex2jar file from that folder: now you get class files.

6. Save all of these class files (In jd-gui, click File -> Save All Sources) by src name. At this stage you get the java source but the .xml files are still unreadable, so continue.

Step 3:

Now open another new folder

1. Put in the .apk file which you want to decode

2. Download the latest version of apktool AND apktool install window (both can be downloaded from the same link) and place them in the same folder

3. Open a command window

4. Now run command like apktool if framework-res.apk (if you don't have it get it here)and next

5. apktool d myApp.apk (where myApp.apk denotes the filename that you want to decode)

now you get a file folder in that folder and can easily read the apk's xml files.

Step 4:

It's not any step, just copy contents of both folders(in this case, both new folders) to the single one

and enjoy the source code...

This is an alternative description - just in case someone got stuck with the description above. Follow the steps:

1. download apktool.bat (or apktool for Linux) and apktool_<version>.jar from http://ibotpeaches.github.io/Apktool/install/
2. rename the jar file from above to apktool.jar and put both files in the same folder

3. open a dos box (cmd.exe) and change into that folder; verify that a Java Environment is installed (for Linux check the notes regarding required libraries as well)
4. Start: apktool decode [apk file]

   Intermediate result: resource files, AndroidManifest.xml

5. unzip APK file with an unpacker of your choice

   Intermediate result: classes.dex

6. download and extract dex2jar-0.0.9.15.zip from http://code.google.com/p/dex2jar/downloads/detail?name=dex2jar-0.0.9.15.zip&can=2&q=
7. drag and drop classes.dex onto dex2jar.bat (or enter <path_to>\dex2jar.bat classes.dex in a DOS box; for Linux use dex2jar.sh)

   Intermediate result: classes_dex2jar.jar

8. unpack classes_dex2jar.jar (might be optional depending on used decompiler)
9. decompile your class files (e.g. with JD-GUI or DJ Decompiler)

   Result: source code

Note: it is not allowed to decompile third party packages; this guide is intended to recover personal source code from an APK file only; finally, the resulting code will most likely be obfuscated

While you may be able to decompile your APK file, you will likely hit one big issue:

it's not going to return the code you wrote. It is instead going to return whatever the compiler inlined, with variables given random names, as well as functions given random names. It could take significantly more time to try to decompile and restore it into the code you had, than it will be to start over.

Sadly, things like this have killed many projects.
For the future, I highly recommend learning a Version Control System, like CVS, SVN and git etc.

and how to back it up.

There is also a new application on the Play Store with which it is possible to decompile an apk (system applications too) and view the source code right on your smartphone. It saves the files to your SD card so you can view it on your computer too. It does not require root or something else.

Just install and have fun. I think this is the easiest way to decompile an app.