Abstract: A line is a transversal of a collection C of k convex sets in R^d
if it stabs all elements in C. The boundary of T(C)- the space of
all line transversals of C- is described by the means of certain manifolds,
each formed by transversals tangent to a single element of C.
When the elements of C are pairwise disjoint, any bound on the complexity
of T(C) is also a bound on the number g_d(k) of so called geometric
permutations, i.e., the maximum number of distinct orders the transversal lines
visit individual elements of C.

We overview the existing upper bound derivations and lower bound constructions
for g_d(k), and describe recent progress in bounding the complexity of T(C)
when C consists of k convex polyhedra with a total of n facets (our joint work
with Haim Kaplan and Micha Sharir). Our analysis (which also works for possibly
intersecting polyhedra!) is curiously connected with geometric permutations
in three dimensions. Finally, we mention some fascinating open problems.