We prove a number of results concerning Armendariz rings and Gaussian rings. Recall that a (commutative) ringRis (Gaussian) Armendariz if for two polynomialsf,g∈R[X] (the ideal ofRgenerated by the coefficients off gis the product of the ideals generated by the coefficients offandg)fg= 0 impliesaibj=0 for each coefficientaioffandbjofg. A number of examples of Armendariz rings are given. We show thatRArmendariz implies thatR[X] is Armendariz and that forRvon Neumann regularRis Armendariz if and only ifRis reduced. We show thatRis Gaussian if and only if each homomorphic image ofRis Armendariz. Characterizations of whenR[X] andR[X] are Gaussian are given.