In a graph $G(V,E)$, a vertex $v$ dominates a vertex $u$ if $u=v$ or there is an edge from $v$ to $u$. A dominating set of $G$ is a subset $D$ of $V$ such that every vertex in $V$ is dominated by at least one vertex in $D$. Domination problem, that is proposed by K{"o}nig, and its variations have fruitful literature more than $300$ publications. One class of those interesting variations is the {it multiple domination problems}, i.e., each vertex in $V$ requires to be dominated by more than one vertex in $D$. In this thesis, we study the $k$-tuple domination problem on several graph classes. We give a linear time algorithm of weighted $k$-tuple domination problem, and prove NP-completeness of $k$-tuple domination problem on some graph classes like planar graphs.