Some galaxies are dubbed ``active'', which means that they radiate much more energy than can be sustained by a normal evolution rate, or average gas consumption rate, given their size and gas reservoir mass. Their activity could be due to a starburst, or to a material-accreting compact nucleus. Both are often related or simultaneous. The activity has a short duration with respect to the galaxy life-time, but can be recurrent. A key issue is to understand the mechanisms that trigger the activity, and how a large amount of material can be funneled to the central regions, to fuel this activity.
A first possibility is provided by the dense star clusters that are present near the nucleus. These nearby stars can provide gas fuel to the nucleus through their mass loss rates, or even the tidal disruption of stars can liberate their whole gas mass. We will present and discuss the various physical phenomena involved, and their respective time-scales in section 2: only those stars that have low angular momentum orbits are available to fuel the activity, but these can be replenished through dynamical diffusion when depleted. A bigger problem arises when the nearby stars are not numerous enough to sustain the activity: mass has then to come from the whole galaxy in a short time-scale, and this rises the problem of angular momentum transfer, that must involve internal or externally-trigerred gravitational instabilities. Note that the mass fueling can occur in two steps: a first instability drives the gas towards the center, giving rise to a nuclear starburst and the formation of dense nuclear star clusters. Then, the active nucleus can be fueled by the evolution of the dense star cluster.
The gravitational instabilities of a galaxy disk are described in the next sections. Two-fluid instabilities are considered, and the critical role of gas is emphasized (section 6). Because of its dissipative character, the gas can cool down as soon as instabilities heat it, and maintain non-axisymmetric features like spiral structure.The corresponding gravity torques are the tool to tranfer angular momentum. The most wide-spread instabilities are the m = 2 spirals and bars: their formation mechanisms, their family of orbits, their resonances, etc... are essential to better understand the dynamical gas fueling (section 4). It will be shown how bars can be destroyed, or re-born, how bars within bars develop, and how material could be driven towards the nucleus more efficiently through a hierarchy of bars.
Also very frequent are the m = 1 instabilities, lopsidedness and off-centring. Several possible mechanisms will be described for these instabilities, that can also favor material infall to the center (section 5). These include counter-rotation, warps or peculiar gaseous instabilities in a near-keplerian disk, a situation common in the neighborhood of central supermassive black holes.
Finally, external triggers through galaxy interactions and mergers should also be emphasized. Observational evidence of internal and external fueling will be discussed and compared. The role of gas and bars in minor and major mergers is investigated through numerical simulations. These external mechanisms are favored by hierarchical scenarios of galaxy formation, where massive galaxies are the result of the merging of several smaller entities. In this frame, it is expected that the central black hole mass increases in parallel to the mass of the galaxy, as observations seem to confirm.
One consequence of galaxy merging is the possibility of binary black hole formation. The evolution and history of the black hole population will be investigated through the main dynamical mechanisms (section 8), and implications on the fueling will be explored. The Starburst-AGN connection (competition or collaboration) is then described, in a cosmological perspective.