The topology of real algebraic varieties is the study of the
topology of objects that can be defined real algebraically and particularly the restrictions
that the real algebraic structure imposes on the topology. Hilbert's famous address motivated
an interest in the so called Hilbert's sixteenth problem of classifying, up to isotopy,
real algebraic curves of a given degree. While successful only up to degree 7, it has
motivated many natural and interesting questions and generalizations. We will look at
some of these questions including a criterion for curves that have the maximum
permissible number of components in terms of the Jacobian of the complexification
of the curve, the reduction of the classification of real planar rational quartics
to the simple combinatorial classification of "extended" chord diagrams, the classification
of real rational knots of low degrees, and an application of braid groups in
constructing real rational representatives of knots.