  We introduce forest diagrams and strand diagrams for elements of Thompson's
group F. A forest diagram is a pair of infinite, bounded binary forests
together with an order-preserving bijection of the leaves. Using forest
diagrams, we derive a simple length formula for elements of F, and we discuss
applications to the geometry of the Cayley graph, including a new upper bound
on the isoperimetric constant (a.k.a. Cheeger constant) of F. Strand diagrams
are similar to tree diagrams, but they can be concatenated like braids.
Motivated by the fact that configuration spaces are classifying spaces for
braid groups, we present a classifying space for F that is the ``configuration
space'' of finitely many points on a line, with the points allowed to split and
merge in pairs. Strand diagrams are related to a description of F as a
groupoid, which we use to derive presentations for F, T, V, and the braided
Thompson group BV.
  In addition to the new results, we include a thorough exposition of the basic
theory of the group F. Highlights include a simplified proof that the
commutator subgroup of F is simple, a discussion of open problems (with a focus
on amenability), and a simplified derivation of the standard presentation and
normal forms for F using forest diagrams.
