Let G be a connected reductive linear algebraic group over the field
Q of rational numbers, K a maximal compact subgroup and Gamma an
arithmetic subgroup of G. Suppose that D=G(R)/K admits a
G(R)-invariant complex structure and that V=Gamma\D has
the structure of a quasi-projective complex algebraic variety. The
special points of V come from the fixed points in D of the maximal
compact tori in G. These points play a key role in many arithmetic
questions, especially when V is a modular variety for families of
abelian varieties. We discuss some recent results about such points, in
particular concerning automorphic functions and monodromy. Some of these
results have their origins in Hilbert's 7th problem, in class field
theory and in questions posed by Siegel on the exceptional sets of
G-functions and modular functions. For example, if F=F(a,b,c,z) is the
Gauss hypergeometric function (an example of a G-function) and a, b,
c are rational, the finiteness or not of certain sets of special points
at which F takes algebraic values determines the arithmetic nature of the
monodromy group of F.