In this paper, we propose nanodot- and nanowire-based metal-oxide-semiconductor field effect transistors (MOSFETs) that can be fabricated by a process that does not require extremely high lithographic resolution. The MOSFET devices remain functional even when nanodots and nanowires of various sizes are randomly distributed. The device physics and modeling of nanodot and nanowire MOSFETs are presented. The (analytical) general solution to Poisson's equation for nanowire MOSFET is given. Moreover, we apply a boundary-condition coupled transformation technique to linearize the original Poisson's equation for nanodot MOSFET to such a form that an analytical solution can be obtained. It is shown that this analytical solution is an accurate description of the electric potential and inversion charge concentration in nanodot MOSFET. The stochastic characteristics of the device parameters of nanodot and nanowire MOSFETs are studied and several important process control issues are discussed. The approximate formulas for calculating the total inversion charge in the ultrathin bodies of nanodot and nanowire MOSFETs are presented.