Let G be a group, and $${{\,\mathrm{Sol}\,}}(G)=\{x \in G : \langle x,y \rangle \text { is solvable for all } y \in G\}$$ . We associate a graph $$\mathcal {NS}_G$$ (called the non-solvable graph of G) with G whose vertex set is $$G \setminus {{\,\mathrm{Sol}\,}}(G)$$ and two distinct vertices are adjacent if they generate a non-solvable subgroup. In this paper, we study many properties of $$\mathcal {NS}_G$$ . In particular, we obtain results on vertex degree, cardinality of vertex degree set, graph realization, domination number, vertex connectivity, independence number and clique number of $$\mathcal {NS}_G$$ . We also consider two groups G and H having isomorphic non-solvable graphs and derive some properties of G and H. Finally, we conclude this paper by showing that $$\mathcal {NS}_G$$ is neither planar, toroidal, double toroidal, triple toroidal nor projective.