I just installed a new 72 inch long Marble vanity in my bathroom. It came with a 72 inch long by 4 inch tall Marble Backsplash. How do I install this backsplash? Do I just put an adhesive behind it? What about the gaps? How do I fill the gap of the top of the backsplash where it hits the wall?

You can use some adhesive to attach it to the wall and fill the gaps along the wall with caulk.

Adhesive caulking. Run a wavy bead the length of back splash. Set in place, caulk joints between the top and the back splash; at back splash and wall.