The progress in computing results in the possibility to compute numerically the orbit of a single celestial object in almost all problems, even when there are close approaches and in other highly unstable cases. This has, however, only moved forward the frontiers of research in Dynamical Astronomy. Now the goal is to study the dynamical behaviour of entire populations of celestial bodies, either real (observed) or only hypothetical (simulated).

The populations discussed in the most recent research include small solar system bodies, like asteroids, comets, meteoroid streams, rings and interplanetary dust particles; stars in aggregations, including clusters and even entire galaxies; artificial objects, like space debris; hypothetical extrasolar planets, in an effort to determine the regions of phase space where such objects can be found; virtual objects, to represent the uncertainty of the orbit as determined from the observations.

The methods developed to cope with population dynamics problems range from the analytical, semianalytical and synthetic theories of proper elements, the improvements in performance and accuracy of numerical integration, the approximate integrators with important qualitative properties (e.g., symplectic methods). Even the availability of efficient computational tools, giving approximate but rich information on a very large number of orbits, solves only half of the problem: how should the results be represented? The availability of advanced computer graphics, including color coded maps and animations, has made possible to present enormous amounts of information in one slide, but this might be dowloading the difficulty of understanding the results on the user.

Let us mention a few examples. We have now a good understanding of the dynamical structure of the asteroid main belt over intermediate time scales (few tens to a few hundreds millions of years). To extend such understanding to the entire history of the solar system we need to account for chaotic diffusion, non gravitational perturbations (especially Yarkovsky effect) and collisional evolution. Asteroid families, formed by catastrophic disruption (and/or craterization) of parent bodies, are now a well established fact. Similarly, the sucess of Leonid predictions over recent years showed that dynamic model of meteoroids streams are reliable, although over shorter time scales. However the very success in creating huge proper element catalogs (with ~100,000 items) has created a difficult situation, in which asteroid families cannot be defined just by statistical methods, as groupings contrasting with a sparse background. Very similar problems arise in the study of meteor showers, again because of the very large data sets now available.

More difficult problems of unstable, strongly chaotic dynamics arise in the study of the transport of meteorites and Near Earth Objects (NEO). Some meteoroid streams, such as the Quadrantid, also exhibit many chaotic properties. The subject of the study is not anymore the single object, but the entire Earth-crossing population. By using very efficient simplectic integrators a large amount of qualitative and quantitative information has been gathered, but then the problem is how to understand and represent the results. Model populations of NEO are the most useful output of these studies. However, the question arises of the sensitivity of these models to assumptions and approximations used: as an example, non gravitational perturbations are relevant in this process, but a quantitative measure of this relevance is not yet well established.

The transport of Trans Neptunian Objects (TNO) and comets to the inner Solar System has also been studied by large scale numerical integrations; but, are the current models fully self consistent, taking into account all the effects, including collisions and physical 'aging' of comets? Interplenetary dust dynamics, including the formation of dust belts, is basically understood, but the relative importance of different sources is still the subject of discussion.

Although stellar dynamics is beyond the scope of this meeting, it would be wise to learn about the methods used in this context, in particular in the methods used to handle stellar close approaches in the modeling of dense clusters. The relationship of the methods used in this context with the ones used, e.g., for planetary encounters of asteroids and comets needs to be discussed.

The dynamics of planetary systems very different from our own has been a subject of intense study in the recent years, because of the discovery of many extrasolar planetary systems with puzzling dynamical structures. While the study of a single known system is always feasible, it is not yet clear how to explore the phase space of the possible systems to constrain the observational searches, classify and understand the new discoveries.

A large populations of artificial celestial bodies in geocentric orbit has been created by the space age. To model space debris, and also the meteoroid background, in a way operationally useful to increase the security of space assets, such as telecommunication satellites and the International Space Station, is a challenge with significant practical, even economic, implication.

In the effort to solve the problem to detect possibilities of future asteroid/comet impacts on our planet it was realized that the present uncertainty on the real orbit of one of these bodies needs to be represented by means of a swarm of Virtual Asteroids (VA). Thus to compute the probability of a future impact we need to solve a population dynamics problem. Advanced methods to solve this problem involve describing the VA by means of a geometric object, either one or two dimensional, rather than using a swarm of independent points.

Although each specialist can tell an interesting tale based on the results of its own solution to the specific population dynamic problem, the main focus of the exchange of information should be on the methods. Indeed the export of methods from one problem to another can be very effective.

This meeting is intended to pursue this cross-fertlization on top of presenting specific new results. Although it is premature to decide the speakers and subject of the invited talks, the format of the meeting should include a significant number of review presentations intended to give to specialists of different populations access to the basic ideas and methods used by others. However, of course there will be room for the presentation of the most recent results, since this field has been advancing very rapidly in the last few years.

The timing of this meeting has been chosen avoding superposition with other related meetings and taking into account that a long time elapsed since the last IAU sponsored meeting not only on this subject, but even connected to the same discipline, namely Celestial Mechanics and Dynamical