The orbital evolution of comets with high values of the Tisserand constant is studied for a time inter-val of 800 years. Scenarios of dynamical evolution are obtained for 98 comets. Particular features of the orbital evolution of the comets of this class are singled out. statistical analysis of all encounters with Jupiter is carried out. Two existing possibilities are analyzed for the behavior of the orbital elements of the comets in the encounter region. The basic characteristics of encounters with Satum and Uranus are presented. comparison is made between encounters with Satum (Uranus) and encounters with Jupiter. Both common and particular features of such encounters are considered. The comets under consideration have orbits with a steadily low inclination, which are tangent to the Jovian orbit at the aphelion or at the perihelion. The eccentric-ity of their orbits does not exceed e = 0.6. Of these com-ets, 81% have a timespan ∆τ in their evolution during which they move in low-eccentricity orbits (l-orbits). Residence in an l-orbit increases the instability of the orbit itself and, evidently, of the comet moving in it as well. The instability is due to very efficient encounters with Jupi-ter which took place during the timespan ∆τ. Such encounters can be accompanied not only by transitions of aphelion to perihelion, but also by multiple minima of the jovicentric-distance function of the comets and by their temporary satellitic capture. This work was supported by RFBR (Grant 06-02-16512).