Abstarct: Consider a connected undirected graph G = (V, E), a subset of vertices C ⊆ V ,and an integer r ≥ 1; for any vertex v ∈ V ,let Br(v) denote the ball of radius r centered at v, i.e., the set of all vertices within distance r from v. If for all vertices v ∈ V (respectively, v ∈ V \C), the sets Br(v) ∩ C are all nonempty and different, then we call C an r-identifying code (respectively, an r-locating-dominating code). We study the smallest cardinalities or densities of these codes in this dissertation. In the first chapter of this dissertation, we state required definitions and theorems of graph theory. In Chapter 2, we describe some properties of translations in Z2, properties which will be necessary to our study of periodic codes in the four grids including square lattice, triangular lattice, king lattice and hexagonal grid for small r. In Chapter 3, We prove here that for all r > 1, D(Gk, r) = 4 1 r. In Chapter 4, we determine the exact value of the smallest possible density of an r-identifying code and r-locating-dominating code in the finite and infinite chains and finite cycles , for all r ≥ 1.