Tihany Abbey is the peninsula's landmark, a Benedictine monastery standing above the lake with twin Baroque towers visible from much of Balaton. The present church is mainly 18th century, but the foundation goes back to the 11th century and King Andrew I, whose tomb is in the crypt.
The interior is handsome, but the setting is what makes the visit memorable. Step outside and the lake opens in two directions, with Balatonfüred to the northeast and the ferry crossing below. It is the one sight in Tihany that everyone visits, so arrive early if you want any quiet.