In this paper, using the method of blow-up analysis, we establish a generalized Trudinger-Moser inequality on a compact Riemannian surface with conical singularity. Precisely, let (Σ,D) be such a surface with divisor $$D=\Sigma_{i=1}^m\beta_{i}p_{i}$$ D = Σ i = 1 m β i p i , where βi > −1 and pi ∈ Σ for i = 1, …, m, and g be a metric representing D. Denote b0 = 4π(1 + min1⩽i⩽mβi). Suppose ψ : Σ → ℝ is a continuous function with ∫Σψdvg ≠ 0 and define $$\lambda _1^{**} (\sum ,g) = \mathop {\inf }\limits_{u \in H^1 (\sum ,g),\smallint _\sum \psi udv_g = 0,\smallint _\sum u^2 dv_g = 1} \int_\sum {\left| {\nabla _g u} \right|^2 dv_g .}$$ λ 1 ∗ ∗ ( ∑ , g ) = inf u ∈ H 1 ( ∑ , g ) , ∫ ∑ ψ u d v g = 0 , ∫ ∑ u 2 d v g = 1 ∫ ∑ | ∇ g u | 2 d v g . Then for any $$\alpha\in[0,\lambda_1^{**}(\Sigma, g))$$ α ∈ [ 0 , λ 1 * * ( Σ , g ) ) , we have $$\mathop {\sup }\limits_{u \in H^1 (\sum ,g),\smallint _\sum \psi u = 0,\smallint _\sum \left| {\nabla _g u} \right|^2 dv_g - \alpha \smallint _\sum u^2 dv_g \leqslant 1} \int_\sum {e^{b_0 u^2 } dv_g < + \infty .}$$ sup u ∈ H 1 ( ∑ , g ) , ∫ ∑ ψ u = 0 , ∫ ∑ | ∇ g u | 2 d v g − α ∫ ∑ u 2 d v g ⩽ 1 ∫ ∑ e b 0 u 2 d v g < + ∞ . When b > b0, the integrals $$\int_\sum {e^{bu^2 } dv_g }$$ ∫ ∑ e b u 2 d v g are still finite, but the supremum is infinity. Moreover, we prove that the extremal function for the above inequality exists.