  Bifurcations can cause dynamical systems with slowly varying parameters to
transition to far-away attractors. The terms ``critical transition'' or
``tipping point'' have been used to describe this situation. Critical
transitions have been observed in an astonishingly diverse set of applications
from ecosystems and climate change to medicine and finance. The main goal of
this paper is to give an overview which standard mathematical theories can be
applied to critical transitions. We shall focus on early-warning signs that
have been suggested to predict critical transitions and point out what
mathematical theory can provide in this context. Starting from classical
bifurcation theory and incorporating multiple time scale dynamics one can give
a detailed analysis of local bifurcations that induce critical transitions. We
suggest that the mathematical theory of fast-slow systems provides a natural
definition of critical transitions. Since noise often plays a crucial role near
critical transitions the next step is to consider stochastic fast-slow systems.
The interplay between sample path techniques, partial differential equations
and random dynamical systems is highlighted. Each viewpoint provides potential
early-warning signs for critical transitions. Since increasing variance has
been suggested as an early-warning sign we examine it in the context of normal
forms analytically, numerically and geometrically; we also consider
autocorrelation numerically. Hence we demonstrate the applicability of
early-warning signs for generic models. We end with suggestions for future
directions of the theory.
