Image: Pennsylvania Department of Conservation and Natural Resources - Forestry Archive, Bugwood.org
Even though this tree is considered to be invasive, most of the web hits I got touted its beauty, quality of the wood, and rapid growth. Many mentioned that it is highly valued in Japan. The fact that it is fast growing makes it attractive for some purposes. It also has attractive flowers, and the leaves are indeed impressive. For a different take, see the Plant Conservation Alliance Alien Plant Working Group fact sheet on Paulownia - it makes their "least wanted" list.
In my book, anything that grows something like 8 feet in 2-3 months is almost by definition invasive. And the damn thing is firmly rooted. I've wrestled it part way out of the ground, but I'm going to need help getting those roots to let go. And it is reputed to sprout easily from roots, so I need to get it out of there.