When designers and builders are unconstrained by convention, they produce motorcycles that set new benchmarks for style and performance.

The motorcycles in this selection come from different parts of the world, belong to a variety of classes, and vary widely in appearance—from each other and from virtually every other model. Indeed, each is sufficiently rare and unusual looking to serve as a statement from and about the rider. Outrageous designs and advanced materials distinguish these bikes, but so…