One- and two-dimensional random walk processes and their effects on EPR (electron spin paramagnetic resonance) line shapes are examined. Discrepancies between the coarse continuum model and the discrete hopping model are discussed. Closed-form formulas are prescribed for the distribution function, correlation (memory) function and the EPR line shapes, covering the entire range between the slow and the fast hopping limits. Deviation from Gaussian or Lorentzian line shape is shown for intermediate hopping rates. Differences between the cases of a closed loop and an open chain are illustrated. Time-domain analysis of line shape measurements can provide information about the hopping rate constants, site numbers in an open or closed chain, and the activation energy.