Enumerative geometry is a classical topic of algebraic geometry with many interesting connections to other parts of mathematics, but also to theoretical physics. Here we are concerned with enumerative geometry of curves on surfaces. A node of a curve is a transversal self-intersection. The Severi degrees count degree d curves with a given number of nodes in the projective plane through the appropriate number of general points, or more generally in a linear system on a projective surface. The Welschinger invariants count the corresponding numbers of real algebraic curves satisfying real point conditions. Both the Severi degrees and the Welschinger invariants can also be computed in tropical geometry. Using tropical geometry we introduce refined Severi degrees, Laurent polynomials in a variable y, which interpolate between the Severi degrees (at y=1) and the Welschinger invariants (at y=1). After reviewing the invariants, we report on recent progress in studying these invariants, e.g. relating them to refined Donaldson-Thomas invariants and progress towards understanding the structure of their generating functions.