Persistant current in isolated mesoscopic rings is studied using the continium and tight-binding models of independent electrons. The calculation is performed with disorder and also at finite temperature. In the absence of disorder and at zero temperature agreement is obtained with earlier results by D. Loss et. al., in that there is half quantum flux periodicity for a large and odd number of electrons, but full quantum periodicity for any even number of electrons in the ring. Strong, disorder converts the period into full quantum periodicity. Finite temperature reduces the magnitude of the current, but preserves the quantum flux periodicity at zero temperature. However the sign of the current may change as disorder or temperature is increased. A generalization of the parity effect, previously discussed by Legett, Loss and Kusmartsev is described for the case where there are electrons with spin, influenced by finite temperature and disorder.

17 pages(LaTeX), 4 figures by request, accepted by Phys. Rev. B and Technical Report of ISSP, Tokyo University