Counting real curves with passage/tangency conditions

We study the following question: given a set ℘ of 3d-2 points and an immersed curve Γ in the real plane ℝ², all in general position, how many real rational plane curves of degree d pass through these points and are tangent to this curve. We count each such curve with a certain sign, and present an explicit formula for their algebraic number. This number does not change under small regular homotopies of a pair (℘, Γ) but jumps (in a well-controlled way) when in the process of homotopy we pass a certain singular discriminant. We discuss the relation of such enumerative problems with finite-type invariants. Our approach is based on maps of configuration spaces and the intersection theory in the spirit of classical algebraic topology.