Summary: We look for solutions to the Schrödinger-Poisson-Slater equation \[- \Delta u + \lambda u - \gamma ( | x |^{- 1} \ast | u |^2 ) u - a | u |^{p - 2} u = 0 \text{ in } \mathbb{R}^3 ,\tag{0.1}\] which satisfy \[ \| u \|_{L^2 ( \mathbb{R}^3 )}^2 = c\] for some prescribed \(c > 0\). Here \(u \in H^1( \mathbb{R}^3)\), \(\gamma \in \mathbb{R}\), \(a \in \mathbb{R}\) and \(p \in(\frac{ 10}{ 3}, 6]\). When \(\gamma > 0\) and \(a > 0\), both in the Sobolev subcritical case \(p \in(\frac{ 10}{ 3}, 6)\) and in the Sobolev critical case \(p = 6\), we show that there exists a \(c_1 > 0\) such that, for any \(c \in(0, c_1)\), (0.1) admits two solutions \(u_c^+\) and \(u_c^-\) which can be characterized respectively as a local minima and as a mountain pass critical point of the associated Energy functional restricted to the norm constraint. In the case \(\gamma > 0\) and \(a < 0\), we show that, for any \(p \in(\frac{ 10}{ 3}, 6]\) and any \(c > 0\), (0.1) admits a solution which is a global minimizer. Finally, in the case \(\gamma < 0, a > 0\) and \(p = 6\) we show that (0.1) does not admit positive solutions.