  Motivated by the study of Fano type varieties we define a new class of log
pairs that we call asymptotically log Fano varieties and strongly
asymptotically log Fano varieties. We study their properties in dimension two
under an additional assumption of log smoothness, and give a complete
classification of two dimensional strongly asymptotically log smooth log Fano
varieties. Based on this classification we formulate an asymptotic logarithmic
version of Calabi's conjecture for del Pezzo surfaces for the existence of
K\"ahler--Einstein edge metrics in this regime. We make some initial progress
towards its proof by demonstrating some existence and non-existence results,
among them a generalization of Matsushima's result on the reductivity of the
automorphism group of the pair, and results on log canonical thresholds of
pairs. One by-product of this study is a new conjectural picture for the small
angle regime and limit which reveals a rich structure in the asymptotic regime,
of which a folklore conjecture concerning the case of a Fano manifold with an
anticanonical divisor is a special case.
