On the arrangement of ovals of plane real algebraic curves

The problem of the relative position of the ovals of non-singular flat projective real algebraic curve constitutes the first part of Hilbert's 16th problem (the second part is about the limit cycles). I will give a general overview of the results and approaches, paying more attention to my results, because I know them better. In particular, I plan to discuss the connection of this problem with its analogue for real (i.e., invariant under the complex conjugation) pseudoholomorphic curves.