A near-ring N is said to be strongly reduced if, for a ${in}$ N, $a^2{in}N_c$ implies $a{in}N_c$, where $N_c$ denotes the constant part of N. We investigate some properties of strongly reduced near-rings and apply those to the study of left strongly regular near-rings. Finally we classify all reduced and strongly reduced near-rings of order ${leq}$ 7 using the description given in J. R. Clay [1].