Summary: We study the uniqueness of positive solutions of the following coupled nonlinear Schrödinger equations: \[ \begin{cases} \Delta u_1 - \lambda_1 u_1 + \mu_1 u_1^3 + \beta u_1 u_2^2 = 0 \quad \text{in} \quad \mathbb R^N,\\ \Delta u_2 - \lambda_2 u_2 + \mu_2 u_2^3 + \beta u_1^2 u_2 =0 \quad \text{in} \quad \mathbb R^N, \\ u_1 > 0, u_2 > 0, u_1, u_2 \in H^1 (\mathbb R^N), \end{cases} \] where \(N \leq 3\), \(\lambda_1, \lambda_2, \mu_1, \mu_2\) are positive constants and \(\beta \geq 0\) is a coupling constant. We prove first the uniqueness of positive solution for sufficiently small \(\beta > 0\). Secondly, assuming that \(\lambda_1 = \lambda_2\), we show that \(u_1 = u_2 \sqrt{\beta - \mu_1}/\sqrt{\beta - \mu_2}\) when \(\beta > \max\{\mu_1, \mu_2\}\) and thus obtain the uniqueness of positive solution using the corresponding result of scalar equation. Finally, for \(N = 1\) and \(\lambda_1 = \lambda_2\), we prove the uniqueness of positive solution when \(0 \leq \beta \notin [\min\{\mu_1, \mu_2\}, \max\{\mu_1, \mu_2\}]\) and thus give a complete classification of positive solutions.