The concept of Sombor index $\left(SO\right)$ was recently introduced by Gutman in the chemical graph theory. It is a vertex-degree-based topological index and is denoted by Sombor index $SO$: $SO=SO\left(G\right)={\displaystyle \sum _{v_i{v}_j\in E\left(G\right)}}\sqrt{d_G{\left(v_i\right)}^2+d_G{\left(v_j\right)}^2},$ where $d_G\left(v_i\right)$ is the degree of vertex $v_i$ in G. Here, we present novel lower and upper bounds on the Sombor index of graphs by using some graph parameters. Moreover, we obtain several relations on Sombor index with the first and second Zagreb indices of graphs. Finally, we give some conclusions and propose future work.