**By Stefano Allesina** (University of Chicago)

Since the work of Robert May in 1972, the local asymptotic stability of
large ecological systems has been a focus of theoretical ecology.Here I
review May's work in the light of Random Matrix Theory, the field of
mathematics devoted to the study of large matrices whose coefficients
are randomly sampled from distributions with given characteristics.
I show how May's celebrated ``stability criterion'' can be derived using
Random Matrix Theory, and how extensions of the so-called circular law
for the limiting distribution of the eigenvalues of large random matrix
can further our understanding of ecological systems.
I conclude by enumerating a number of challenges, whose solution is
going to greatly improve our ability to understand the dynamics of large
cological networks.