  We introduce and begin the topological study of real rational plane curves,
all of whose inflection points are real. The existence of such curves is a
corollary of results in the real Schubert calculus, and their study has
consequences for the important Shapiro and Shapiro conjecture in the real
Schubert calculus. We establish restrictions on the number of real nodes of
such curves and construct curves realizing the extreme numbers of real nodes.
These constructions imply the existence of real solutions to some problems in
the Schubert calculus. We conclude with a discussion of maximally inflected
curves of low degree.
