  The stability of ecological systems is a fundamental concept in ecology,
which offers profound insights into species coexistence, biodiversity, and
community persistence. In this article, we provide a systematic and
comprehensive review on the theoretical frameworks for analyzing the stability
of ecological systems. Notably, we survey various stability notions, including
linear stability, sign stability, diagonal stability, D-stability, total
stability, sector stability, structural stability, and higher-order stability.
For each of these stability notions, we examine necessary or sufficient
conditions for achieving such stability and demonstrate the intricate interplay
of these conditions on the network structures of ecological systems. Finally,
we explore the future prospects of these stability notions.
