We present the first measurements of the kinetics of random motion of individual monomers within large polymer coils. We use double- and single-stranded DNA (dsDNA and ssDNA) as models of semiflexible and flexible polymers, respectively. Fluorescence fluctuations of DNA fragments labeled specifically at a single position reveal the time dependence of the DNA monomer's mean-square displacement <r(2)(t)>. The monomer motions within dsDNA and ssDNA coils are characterized by two qualitatively different kinetic regimes: close to <r(2)(t)> proportional to t(2/3) for ssDNA and <r(2)(t)> proportional to sqrt[t] for dsDNA. While the kinetic behavior of ssDNA is consistent with the generally accepted Zimm theory of polymer dynamics, the kinetic behavior of dsDNA monomers is in good agreement with the Rouse model.