Among the most romantic things to do in Heidelberg is stroll along the well-marked Philosophenweg, or Philosophers' Way.
On the north side of the Neckar, this wonderful pathway runs along the side of Heiligenberg, or Saints' Mountain, and is where the university's philosophers would walk and carry on discussions as they admired the scenery.
It's not hard to see the attraction: the views across the Neckar to the old town are beautiful, and the riverside creates a micr0climate where plants from more southern climates flourish, among them Japanese cherries, cypresses, and even lemons.