  This thesis intends to make a contribution to the theories of algebraic
cycles and moduli spaces over the real numbers. In the study of the
subvarieties of a projective algebraic variety, smooth over the field of real
numbers, the cycle class map between the Chow ring and the equivariant
cohomology ring plays an important role. The image of the cycle class map
remains difficult to describe in general; we study this group in detail in the
case of real abelian varieties. To do so, we construct integral Fourier
transforms on Chow rings of abelian varieties over any field. They allow us to
prove the integral Hodge conjecture for one-cycles on complex Jacobian
varieties, and the real integral Hodge conjecture modulo torsion for real
abelian threefolds.
  For the theory of real algebraic cycles, and for several other purposes in
real algebraic geometry, it is useful to have moduli spaces of real varieties
to our disposal. Insight in the topology of a real moduli space provides
insight in the geometry of a real variety that defines a point in it, and the
other way around. In the moduli space of real abelian varieties, as well as in
the Torelli locus contained in it, we prove density of the set of moduli points
attached to abelian varieties containing an abelian subvariety of fixed
dimension. Moreover, we provide the moduli space of stable real binary quintics
with a hyperbolic orbifold structure, compatible with the period map on the
locus of smooth quintics. This identifies the moduli space of stable real
binary quintics with a non-arithmetic ball quotient.
