Kastamonu's castle, lording it up on a rock outcrop above the central city, is currently undergoing a massive restoration not likely to be finished until at least mid-2020. Although you can't enter, you can still climb up the hill for the view over town. One block south of the Archaeology Museum, turn right onto Åžeyh Åžaban Veli Caddesi and follow the road up.
Parts of the building date from Byzantine times, but most belong to Seljuk and Ottoman reconstructions.