In this paper we extend the NeutroAlgebra & AntiAlgebra to the geometric field, by founding now the NeutroGeometry & AntiGeometry that include the Non-Euclidean Geometries as subclasses. In the second part, we recall the evolution from Paradoxism to Neutrosophy, then to NeutroAlgebra & AntiAlgebra, and afterwards to NeutroGeometry & AntiGeometry, and in general to NeutroStructure & AntiStructure that naturally arise in any field of knowledge. Why all these? Because in our real world the spaces, the elements that populate them and the rules that apply on them are not perfect, not uniform, not complete, they have unclear and conflicting information and do not apply in the same degree to everyone. At the end, we present applications of many NeutroStructures in our real world.