5 Best Sights in Castile–Leon and Castile–La Mancha, Spain

As if Cuenca's famous Casas Colgadas, suspended impossibly over the cliffs below, were not eye-popping enough, they also house one of Spain's finest museums, the Museo de Arte Abstracto Español (Museum of Spanish Abstract Art)—not to be confused with the adjacent Museo Municipal de Arte Moderno (Municipal Museum of Modern Art). Projecting over the town's eastern precipice, these houses originally formed a 15th-century palace, which later served as a town hall before falling into disrepair in the 19th century. In 1927 the cantilevered balconies were rebuilt, and in 1966 the painter Fernando Zóbel created the world's first museum devoted exclusively to abstract art. The works he gathered—by such renowned names as Carlos Saura, Eduardo Chillida, Lucio Muñoz, and Antoni Tàpies—are primarily by exiled Spanish artists who grew up under Franco's regime. The museum has free smartphone audio guides that can be downloaded from the website.

If you don't have a fear of heights, cross this narrow 16th-century footbridge to take in the vertiginous view of the river and equally thrilling panorama of the Casas Colgadas. It's by far the best view of the city. If you've read the popular English novel Winter in Madrid by C. J. Sansom, you'll recognize this bridge from the final scene.

Built in the 12th century atop ruins of a conquered mosque, the cathedral, which looms over the Plaza Mayor, lost its Gothic character in the Renaissance. Inside are the tombs of the cathedral's founding bishops, an impressive portico of the Apostles, and a Byzantine reliquary. There's also a museum in the once-cellar of the Bishop's Palace containing a jewel-encrusted Byzantine diptych of the 13th century, a Crucifixion by the 15th-century Flemish artist Gerard David, a variety of carpets from the 16th through 18th centuries, and two small El Grecos. An excellent audio guide is included in the price of admission.

Not an "Enchanted City" as its name implies, Ciudad Encantada—situated 35 km (22 miles) north of Cuenca—is a series of large, fantastic mushroomlike rock formations erupting in a landscape of pines. It was formed over thousands of years by the forces of water and wind on limestone rocks, and you can see it in under two hours. See if you can spot formations named Cara (face), Puente (bridge), Amantes (lovers), and Olas en el Mar (waves in the sea). Rent a car to get here, or arrange a visit with Ecotourism Cuenca ( www.ecoturismocuenca.com,   64/569–4393).

In what were once the cellars of the Bishop's Palace, this museum's collection includes a jewel-encrusted Byzantine diptych of the 13th century, a Crucifixion by the 15th-century Flemish artist Gerard David, a variety of carpets from the 16th through 18th centuries, and two small El Grecos.