  We provide a concise presentation of the Smaller (SALI) and the Generalized
Alignment Index (GALI) methods of chaos detection. These are efficient chaos
indicators based on the evolution of two or more, initially distinct, deviation
vectors from the studied orbit. After explaining the motivation behind the
introduction of these indices, we sum up the behaviors they exhibit for regular
and chaotic motion, as well as for stable and unstable periodic orbits,
focusing mainly on finite-dimensional conservative systems: autonomous
Hamiltonian models and symplectic maps. We emphasize the advantages of these
methods in studying the global dynamics of a system, as well as their ability
to identify regular motion on low dimensional tori. Finally we discuss several
applications of these indices to problems originating from different scientific
fields like celestial mechanics, galactic dynamics, accelerator physics and
condensed matter physics.
