To start, in iOS 7, Apple introduced the ability to natively zip / unzip files. You also have the option so send the zip files through Mail, Messages, Airdrop and "Open in". The key is: The zip file has to be supported by iOS Some are, and some are not. The first step: Find out if your file is supported. Do this by a simple check of your newly saved file. Try to open it. If it is stuck of "Waiting..." it is probably not supported. Try a different library or use a workaround if this is the case.
Now, doing this programmatically Currently requires the use of a third party library to save zip files and extract them. This is not ideal, since a lot of people / companies avoid using them. That being said, the answer marked correct, ZipArchive is a great third party tool, especially since it now supports Swift. I would recommend using it until Apple introduces a native library.