For a set of 11,900 F stars that have uvby photometry and are in the Hipparcos catalog, we have found 1980 X-ray counterparts (~17%) in the ROSAT All-Sky Survey (RASS). In this paper we analyze X-ray properties of these ROSAT F stars and compare that sample with the entire set of our F stars. The comparison has shown that the ROSAT F stars are on average younger than stars in the whole set, consistent with previous studies that indicate lower X-ray emission in older stars. However, for the ROSAT stars only, the age-X-ray activity relation has proved to be more complicated, with a group of strong X-ray emitters being the oldest rather than the youngest stars. That group determines the shape of the overall relationship between the kinematics and X-ray luminosity. The relationship is in the sense that the mean tangential velocity and velocity dispersion are higher for stars that are more luminous at the X-ray wavelengths, implying that these stars are on average the oldest rather than the youngest. The group of strongest X-ray emitters, log LX > 30.4, which constitutes only a small fraction of the sample stars, ~2.8%, has been found to include a variety of stellar types. It is dominated by very young objects, followed by RS CVn-type stars and other active binaries. As to the sample as a whole, it seems to consist of at least three distinct groups of F stars that differ in their X-ray luminosity and evolution of their X-ray emission. The first one is represented by normal zero-age main-sequence (ZAMS) and post-ZAMS stars whose X-ray emission declines with age, just as expected in the conventional picture of stellar X-ray evolution (roughly half of the sample). The second group consists of strong X-ray emitters that appear to be very young, possibly pre-main-sequence stars. These stars are characterized by high metallicity, high temperature, and high overluminosity (the latter indicating that the star's optical luminosity is too high for the star's temperature and surface gravity). Many of them are located in the distance range ~130-200 pc, near the known regions of ongoing star formation or OB associations. The third group also consists of strong X-ray emitters. However, those are very old rather than very young stars, and they happen to be extremely overluminous as well. Their strong X-ray emission implies that many F stars are capable of increasing rather than decreasing their X-ray luminosity as they get older, thus challenging the idea that X-ray emission continuously declines with age. Most of these old X-ray emitters have significantly evolved away from the ZAMS. If they increase their outer convective zone as they evolve, they would probably be able to maintain a high level of coronal activity, and hence enhanced X-ray emission, despite decaying rotation. Other relationships found for the ROSAT F stars include the correlation of X-ray luminosity with hardness ratio, optical absolute magnitude, effective temperature, overluminosity, metallicity, and reddening. There is also a difference between binary and single stars, with X-ray luminosity being on average slightly higher for unresolved binaries. There is a distinct difference between the groups of soft and hard X-ray emitters in our sample: while the soft sources show a significant correlation between the X-ray luminosity and hardness ratio, no such correlation is seen for hard sources. The tendency for X-ray luminosity to be higher for harder sources implies that stars more luminous in the X-rays have a hotter corona. This tendency breaks up at large hardness values, which may be due to coronal saturation.