  The space of unordered configurations of distinct points in the plane is
aspherical, with Artin's braid group as its fundamental group. Remarkably
enough, the space of ordered configurations of distinct points on the real
projective line, modulo projective equivalence, has a natural compactification
(as a space of equivalence classes of trees) which is also (by a theorem of
Davis, Januszkiewicz, and Scott) aspherical. The classical braid groups are
ubiquitous in modern mathematics, with applications from the theory of operads
to the study of the Galois group of the rationals. The fundamental groups of
these new configuration spaces are not braid groups, but they have many similar
formal properties.
