If an update of the module for which your Marketplace account is enabled is available, you can proceed by clicking “Update” in the list of Prestashop modules.

If you have a ZIP file containing an updated version, you can upload it through the Prestashop “Add new module” function (NEVER UNINSTALL THE PREVIOUS VERSION to not lose all the configurations already made).

After checking the successful update (Prestashop will issue confirmation) perform the following steps to ensure that everything is working properly
Here we report an example with the FastBay module:

Check that the module version (in Modules -> Fastbay example) is the correct one (updated)
Clear the Prestashop cache from Advanced Parameters -> Performance
Clear / clear your browser cache to clear any outdated content
Enter the module configuration and save the configurations (first tab) to complete the update.