A: The best option would be to use a natural stone cleaner mixed in with water to clean your travertine. You can check at your local hardware store which one is more suitable for your project. For deeper cleaning there are the following options: either use boiling water, or another trickier option is to use a water and white vinegar mix.
For the latter you will need to do a spot test since travertine reacts to acidic liquids. If it doesn't etch the travertine you should use it in between the joints of the tiles. Lastly, make sure you have proper drainage in order to prevent moss from coming back too often.