Star architects are descending on the world's museums, and as these world-renowned designers unveil their visions for a wave of new modern cultural institutions, they are entering uncharted territory—stylistically, often, and literally, too.

Daniel Libeskind is finishing his first building in mainland China with the Zhang Zhidong and Modern Industrial Museum in Wuhan. This year's Pritzker Prize winner, Shigeru Ban, just completed his first permanent U.S. museum with the newly opened Aspen Art Museum in Aspen, Colo. Tadao...