Given a graph $G=(V,E)$, a set $B\subseteq V(G)$ is a packing in $G$ if the closed neighborhoods of every pair of distinct vertices in $B$ are pairwise disjoint. The packing number $\rho(G)$ of $G$ is the maximum cardinality of a packing in $G$. Similarly, open packing sets and open packing number are defined for a graph $G$ by using open neighborhoods instead of closed ones. We give several results concerning the (open) packing number of graphs in this paper. For instance, several bounds on these packing parameters along with some Nordhaus-Gaddum inequalities are given. We characterize all graphs with equal packing and independence numbers and give the characterization of all graphs for which the packing number is equal to the independence number minus one. In addition, due to the \begin{color}{red}close\end{color} connection between the open packing and total domination numbers, we prove a new upper bound on the total domination number $\gamma_{t}(T)$ for a tree $T$ of order $n\geq 2$ improving the upper bound $\gamma_{t}(T)\leq(n+s)/2$ given by Chellali and Haynes in 2004, in which $s$ is the number of support vertices of $T$.
