An update from April 2018 is given by Patrick Speissegger.
The idea, going back to Poincaré, is to reduce the two-dimensional counting problem (counting limit cycles in the plane) to a one-dimensional counting problem (counting certain points on a line). Roussarie (1998) showed that Hilbert’s 16th problem follows if a certain "finite cyclicity conjecture" holds. A tameness condition called "o-minimality" allows to reformulate Roussarie's conjecture as a conjecture of o-minimality. Speissegger discusses special cases where o-minimality can be proven and proposes this approach as a promising way to prove Hilbert's 16th problem.