Summary: We consider the following Kirchhoff-Choquard equation \[- M(\| \nabla u \|_{L^2}^2) \Delta u = \lambda f(x) | u |^{q - 2} u + \left(\int_\Omega \frac{| u(y) |^{2_\mu^\ast}}{| x - y |^\mu} d y\right) | u |^{2_\mu^\ast - 2} u \quad\text{in } \Omega,\quad u = 0 \quad\text{on } \partial \Omega,\] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N(N \geq 3)\) with \(C^2\) boundary, \(2_\mu^\ast = \frac{2 N - \mu}{N - 2}\), \(1 < q \leq 2\), and \(f\) is a continuous real valued sign changing function. When \(1 < q < 2\), using the method of Nehari manifold and Concentration-compactness Lemma, we prove the existence and multiplicity of positive solutions of the above problem. We also prove the existence of a positive solution when \(q = 2\) using the Mountain Pass Lemma.