Some important considerations:
1. If you own the Project Pro for Office 365 the update will be simpler - it's like any other update that happens with click to run.
Note: Some organizations can control when updates will be available to your users - so you may not be able to upgrade so quickly or without the help of your IT team.
2. If you chose the Project Pro for Office 365 but is still using Project Server 2013 there will be no problems. The 2013 and 2016 versions are compatible! That fact makes us very happy! You can share .mpp files between versions 2013 and 2016 normally without corrupting files, but with a single point of attention: multiple timelines, which are new features of version 2016, will not be visible when opening a .mpp file in a 2013 version of Project, if that file has multiple timelines.
If you haven't seen this new attribute - here's an example:

3. You cannot have the two versions of Project installed on your computer if you are using click to run installation in version 2013 - so to update by this means you will need to uninstall 2013.
4. You can mix different versions of the Project if version 2013 is installed by msi - so you can run Project Professional 2013 normally with Microsoft Office 365 ProPlus (level '16' - click to run) containing Project 16.
5. You will not be able to have the two versions of Project installed on your computer if you are using click to run installation in version 2013 - so to update by this means you will need to uninstall the products 2013.
How do I know I'm running click to run or msi applications?
Click to run will look like the image below in the Product Information part with the "Updating Options" button:

The msi installed Project will look like this:

The compatibility of 2016 for 2013 is in operation "for the time being" - it occurred specifically thinking about the users of Project Online, where there will be support for the use of Project Professional 2013 connected to Project Online until the 2nd Quarter of 2016.
Original content from the MLPro editorial library, fully preserved and translated for the English website.