A complete list of combinations of the rates of asteroid perihelia and nodes and the corresponding fundamental frequencies of planets, giving rise to secular resonances and involving up to 4 frequencies, is known from the previous work, while for the resonances with 6 frequencies a systematically derived comprehensive list is given here for the first time. There are 28 divisors in the theory of degree up to 4, not all of which can give rise to resonances, while at degree 6 there are (at least) 33 such possibly resonant frequency combinations.

Mapping the secular resonances by plotting the resonant lines in the phase space of proper elements or of secular frequencies, possibly also against the background of known asteroids, enables to straightforwardly identify resonances causing large long periodic variations of asteroid orbital elements, resonances that interact with known families, those that bound the dynamically distinct regions, deplete or disturb asteroids in these regions, etc.

We search for possible differences in rotational frequencies, diameters, albedos, and orbital parameters between Trojans belonging to the L4 and L5 swarms using our own observations and literature data. With increasing number of observational data it becomes evident that the L4 and L5 populations have very similar distributions of most parameters with an exception of orbital inclination distribution.

The history of asteroid families, from their discovery back in 1918, until the present time, is briefly reviewed. Two threads have been followed: on the development of the theories of asteroid motion and the computation of proper elements, and on the methods of classification themselves. Three distinct periods can be distinguished: the first one until mid-1930s, devoted to discovery and first attempts towards understanding of the properties of families; the second one, until early 1980s, characterized by a growing understanding of their importance as key evidence of the collisional evolution; the third one, characterized by an explosion of work and results, comprises the contemporary era. An assessment is given of the state-of-the-art and possible directions for the future effort, focusing on the dynamical studies, and on improvements of classification methods to cope with ever increasing data set.

We consider the role of the dwarf planet Ceres on the secular dynamics of the asteroid main belt. Specifically, we examine the post impact evolution of asteroid families due to the interaction of their members with the linear nodal secular resonance with Ceres. First, we find the location of this resonance and identify which asteroid families are crossed by its path. Next, we summarize our results for three asteroid families, namely (1726) Hoffmeister, (1128) Astrid and (1521) Seinajoki which have irregular distributions of their members in the proper elements space, indicative of the effect of the resonance. We confirm this by performing a set of numerical simulations, showcasing that the perturbing action of Ceres through its linear nodal secular resonance is essential to reproduce the actual shape of the families.

We have recently proposed a new approach to the asteroid family classification by combining the classical HCM method with an automated procedure to add newly discovered members to existing families. This approach is specifically intended to cope with ever increasing asteroid data sets, and consists of several steps to segment the problem and handle the very large amount of data in an efficient and accurate manner. We briefly present all these steps and show the results from three subsequent updates making use of only the automated step of attributing the newly numbered asteroids to the known families. We describe the changes of the individual families membership, as well as the evolution of the classification due to the newly added intersections between the families, resolved candidate family mergers, and emergence of the new candidates for the mergers. We thus demonstrate how by the new approach the asteroid family classification becomes stable in general terms (converging towards a permanent list of confirmed families), and in the same time evolving in details (to account for the newly discovered asteroids) at each update.

In recent years, a rare class of asteroids has been discovered by Cellino et al. (2006), with its distinguishing characteristic being the anomalous polarimetric properties of its members. Named Barbarians, after (234) Barbara, the prototype of the class, these asteroids show negative polarization at unusually high phase-angles compared to normal asteroids. Motivated by the fact that some of the few discovered Barbarians seemed to be related to the Watsonia asteroid family, Cellino et al. (2014) performed a search for more Barbarians among its members. A positive result of this search led to the conclusion that Watsonia is indeed an important repository of Barbarian asteroids. Based on these findings, we decided to analyze this family in detail.

To study an asteroid family it is crucial to determine reliably the list of its members, i.e. to reduce the number of interlopers as much as possible. However, as the number of known asteroids increases fast it becomes more and more difficult to obtain robust list of members of an asteroid family. To cope with these challenges we are proposing a new approach that may help to significantly reduce presence of interlopers among the family members.