8 Summary

Cool stars with external convection envelopes exhibit various magnetic phenomena similar to those observed on the Sun. Starspots are the best studied proxy of stellar magnetism and, thus, represent a key tool for understanding the stellar dynamo. A remarkable progress in observational facilities and numerical techniques for studying starspots achieved for the last two decades allows for a deep insight into the nature of starspots and underlying magnetic fields. In addition to traditional photometric and spectroscopic observations, advanced instrumentation for high-precision spectropolarimetry and (spectro-)interferometry together with foreseen space missions constitute a powerful arsenal for studying starspots and promise for ground-braking discoveries in stellar magnetism.

Starspot properties are revealed with the help of various numerical techniques, such as light-curve modelling and inversions, Doppler and Zeeman-Doppler imaging, molecular line diagnostics, asteroseismology, etc. Parallel brightness and colour variations, i.e., the faintest state being accompanied by the reddest colour, imply that starspots are significantly cooler than the unspotted photosphere, on average by $500 K$ to $2000 K$ . Large variations in brightness, up to $0.6 mag$ , indicate that the spotted areas are huge compared to that of sunspots, up to 30% of the whole stellar surface. Such large cool spots when passing over stellar disks cause strong line profile distortions in the spectra of rapidly rotating stars. Analysis of a time series of spectral line profile variations using Doppler imaging techniques provides the spot distribution over the visible stellar surface. Doppler images reveal that spots on cool rapidly rotating stars are preferably formed at higher latitudes, from $30o$ up to the visible pole, in contrast to sunspots which on average appear at latitudes below $30o$ . Magnetic field measurements suggest that the large starspots represent active regions consisting of smaller, mixed polarity spots.

Summarising the discussion on starspot evolution, the following main activity patterns appear to be common on cool active stars, namely components of binary systems, young single dwarfs and single rapidly rotating giants:

total spottedness variations indicating sunspot-like cycles;
persistent active longitudes $180o$ apart implying non-axisymmetric magnetic fields;
switching of dominant activity between active longitudes: flip-flop cycles;
migration of active longitudes;
differential rotation and butterfly diagrams;
orbital period variations in binaries: changes of the distribution of magnetic fields in the convection envelope.

Non-axisymmetric large-scale magnetic fields are persistent in various types of active stars including the Sun. Their continuous longitude migration implies the presence of differential rotation and provides an opportunity for studying stellar butterfly diagrams. Activity cycles revealed in variations of spot and plage area appear to accompany flip-flop cycles, i.e., periodic switching of dominant activity between opposite longitudes. An analysis of these phenomena can help to identify underlying dynamo modes.

These activity patterns challenge the current stellar dynamo theory. It appears that non-linear dynamo models can qualitatively explain most of the phenomena, although many parameters of the models are not realistic as yet.

Comparing the observed properties of the magnetic field on the Sun and active stars, we can conclude that the activity patterns emphasised above for single stars are also relevant for the solar case. There appears to be more similar between different types of stars than it was thought previously. Thus, fine details of the stellar dynamo can be deduced by studying the Sun, while its global parameters, on an evolutionary time scale, are provided by a sample of active stars.