Four distinct types of asymptotic systolic estimates have been studied recently.

In the case of Riemann surfaces:

1. Upper bounds for the systolic ratio of Riemann surfaces of genus g as g becomes unbounded, by M. Katz and S. Sabourau.

2. Lower bounds for the systole of arithmetic hyperbolic surfaces of genus g as g becomes unbounded, by M. Katz, M. Schaps, and U. Vishne.

For higher dimensional manifolds:

3. Lower bounds for the conformal 2-systole of 4-manifolds as the second Betti number becomes unbounded, by M. Katz and M. Hamilton.

4. Lower bounds for the 1-systole of connected sums of k copies of a
given manifold, as k becomes unbounded, by I. Babenko and
F. Balacheff.