  This thesis is a study of large sets of unit vectors in $\cx^n$ such that the
absolute value of their standard inner products takes on only a small number of
values.
  We begin with bounds: what is the maximal size of a set of lines with only a
given set of angles? We rederive a series of upper bounds originally due to
Delsarte, Goethals and Seidel, but in a novel way using only zonal polynomials
and linear algebra. In the process we get some new results about complex
$t$-designs and also some new characterizations of tightness.
  Next we consider constructions. We describe some generic constructions using
linear codes and Cayley graphs, and then move to two specific instances of the
problem: mutually unbiased bases and equiangular lines. Both cases are
motivated by problems in quantum computing, although they have applications in
digital communications as well.
  Mutually unbiased bases are collections of orthonormal bases with a constant
angle between vectors from different bases. We construct some maximal sets in
prime-power dimensions, originally due to Calderbank, Cameron, Kantor and
Seidel, but again in a novel way using relative difference sets or
distance-regular antipodal covers. We also detail their numerous relations to
other combinatorial objects, including symplectic spreads, orthogonal
decompositions of Lie algebras, and spin models. Peripherally, we discuss
mutually unbiased bases in small dimensions that are not prime powers and in
real vector spaces.
  Equiangular lines are collections of vectors with only one angle between
them. We use difference sets from finite geometry to construct equiangular
lines: these sets do not have maximal size, but they are maximal with respect
to having all entries of the same absolute value. We also include some negative
results about constructions of maximal sets in large dimensions.
