Abstarct: As a generalization of rings, the theory of near-rings ‎has‎ attracted much attention from researchers in past decades. In this thesis, we are interested to study some properties of zero-symmetric near-ring of polynomials and zero-symmetric near-ring of power series. In fact, we want to investigate‏ ‎the ‎structure ‎of‎ some type of elements such as invertible, idempotent, regular, nilpotent, π-regular, ‎clean ‎and ‎zero-divisor elements ‎of ‎the ‎near-ring‎ R_0 [x]‎. Then ‎we ‎determine ‎the ‎structure ‎of ‎these ‎elements ‎in ‎‎‎the skew zero-symmetric near-ring of polynomials ‎R_0 [x;α,δ] ‎‏ ‎and ‎the skew zero-symmetric ‎near-ring ‎of ‎formal ‎power ‎series‎ R_0 [[x;α]]‏. ‎ ‎W‎e are also interested to study some ‎radical-‎theoretical properties of ‎‏‎the near-ring ‎R_0 [x]. In particular, we peruse the quasi-radical of R_0 [x] and determine the ‎relationship ‎between it and ‎the ‎intersection ‎of ‎all ‎maximal ‎left ‎ideals of R_0 [x]. Moreover, we investigate the interplay between the algebraic properties of near-rings and graph-theoretical properties of the assigned (compressed) zero-divisor graph. In fact, we study the ‎diameter ‎of ‎the ‎‎zero-divisor ‎graphs‎ Γ(R_0 [x]),‎ ‎ Γ(R_0 [x;α,δ])‎‏ ‎and‎ Γ(R_0 [[x;α]]),‎ ‎and ‎the ‎compressed ‎zero-divisor ‎graphs‎ Γ_E (R_0 [x] )‎ ‎and‎ Γ_E (R_0 [[x]] ), ‎and ‎give a ‎‎complete characterization for the possible diameters of ‎these ‎graphs.‎‎‎