axiomatic convexity space, introduced by kay and womble [22] , will be the main topic discussed in this thesis. an axiomatic convexity space (x,c), which is domain finite and has regular straight segments, is called a basic convexity space, a weak complete basic convexity space is a basic convexity space which is complete and has c-isomorphic property. if in addition, it is join-hull commutative then it is called (strong) complete basic convexity space. the main results presented are: a generalized line space is a weak complete basic convexity space, a complete basic convexity space is equivalent to a line space; and a complete basic convexity space whose dimension is greater than two or desarguesian and of dimension two, is a linearly open convex subset of a real affine space. finally, we develop a linearization theory by following an approach given by bennett [3]. a basic convexity space whose dimension is greater than two, which is join-hull commutative and has a parallelism property, is an affine space. it can be made into a vector space over an ordered division ring and the members of c are precisely the convex subsets of the vector space.