For an ordered set $W=\{w_1,w_2,\ldots,w_k\}$ of vertices and a vertex $v$ in a connected graph $G$ , the ordered $k$-vector $$r(v|W):=(d(v,w_1),d(v,w_2),\ldots,d(v,w_k))$$ is called the metric representation of $v$ with respect to $W$ , where $d(x,y)$ is the distance between the vertices $x$ and $y$ . The set $W$ is called a resolving set for $G$ if distinct vertices of $G$ have distinct representations with respect to $W$ . A resolving set for $G$ with minimum cardinality is called a basis of $G$ and its cardinality is the metric dimension of $G$ . This concept was introduced in $1975$ by slater . He described the usefulness of these concepts when working with U.S . Sonar and Coast Guard Loran stations . After that it was studied in several papers . The concept of a resolving set has various applications in diverse areas including coin weighing problems , network discovery and verification , robot navigation , mastermind game , problems of pattern recognition and image processing , and combinatorial search and optimization . This thesis is aimed to study the metric dimension of graphs and open problems about it . The metric dimension of the cartesian product of graphs was studied in $2007$ . In this thesis , the metric dimension of lexicographic is considered . Randomly $k$-dimensional graphs are graphs that every $k$-subset of their vertices forms a metric basis . Properties of these graphs are studied and all randomly $k$-dimensional graphs are characterized . In fact , randomly $k$-dimensional graphs have the most number of bases in the other point there are graphs with a unique metric basis . The properties of these graphs are studied in this thesis . In order to characterizing graphs with certain metric dimension , all $n$-vertex graphs with metric dimension $n-3$ are characterized .