We use the method of moments to establish the limiting spectral distribution (LSD) of appropriately scaled large dimensional
random symmetric circulant, reverse circulant, Toeplitz and Hankel matrices which have suitable band structures. The input
sequence used to construct these matrices is assumed to be either i.i.d. with mean zero and variance one or independent and
appropriate finite fourth moment. The class of LSD includes the normal and the symmetrized square root of chi-square with
two degrees of freedom. In several other cases, explicit forms of the limit do not seem to be obtainable but the limits can
be shown to be symmetric and their second and the fourth moments can be calculated with some effort. Simulations suggest some
further properties of the limits.