Enumerative real algebraic geometry asks questions about the real solutions to problems from enumerative geometry, when the geometric conditions in the problem are real. This is the geometric version of the important problem of determining the real solutions to a system of polynomial equations. While this subject is fairly new (the motivating question was asked in 1984, and the first problems were studied from this perspective in the early 1990's) some basic themes are emerging.
One theme is that it seems to be always possible to arrange that the solution figures to an `honest' intersection problem are all real. When this occurs, we say that the enumerative problem is fully real. Another theme is that sometimes there exist lower bounds on the number of real solutions better than trivial bounds. What is not yet clear is whether these lower bounds are sharp, or how widespread is the phenomenon of full reality.
In this survey, we first discuss the situation for real solutions to sparse polynomial systems; it is very illuminating to compare this with what is known about enumerative real algebraic geometry. We continue with a discussion of many enumerative problems which have been studied from the perspective of the real numbers.