  This paper is an overview of our works which are related to investigations of
the integrability of natural Hamiltonian systems with homogeneous potentials
and Newton's equations with homogeneous velocity independent forces. The two
types of integrability obstructions for these systems are presented. The first,
local ones, are related to the analysis of the differential Galois group of
variational equations along a non-equilibrium particular solution. The second,
global ones, are obtained from the simultaneous analysis of variational
equations related to all particular solutions belonging to a certain class. The
marriage of these two types of the integrability obstructions enables to
realise the classification programme of all integrable homogeneous systems. The
main steps of the integrability analysis for systems with two and more degrees
of freedom as well as new integrable systems are shown.
