Let $d_G(v)$ be the degree of the vertex $v$ in a graph $G$. The Sombor index of $G$ is defined as $SO(G) =\sum_{uv\in E(G)}\sqrt{d^2_G(u)+d^2_G(v)}$, which is a new degree-based topological index introduced by Gutman. Let $\mathscr{T}_{n,Δ}$ and $\mathscr{U}_{n,Δ}$ be the set of trees and unicyclic graphs with $n$ vertices and maximum degree $Δ$, respectively. In this paper, the tree and the unicyclic graph with minimum Sombor index among $\mathscr{T}_{n,Δ}$ and $\mathscr{U}_{n,Δ}$ are characterized.