The purpose of this paper is to investigate general mean-field backward stochastic differential equations (MFBSDEs) in multi-dimension with diagonally quadratic generators $f(ω,t,y,z,μ)$, that is, the coefficients depend not only on the solution processes $(Y,Z)$, but also on their law $\mathbb{P}_{(Y,Z)}$, as well as have a diagonally quadratic growth in $Z$ and super-linear growth (or even a quadratic growth) in the law of $Z$ which is totally new. We start by establishing through a fixed point theorem the existence and the uniqueness of local solutions in the ``Markovian case'' $f(t,Y_{t},Z_{t},\mathbb{P}_{(Y_{t},Z_{t})})$ when the terminal value is bounded. Afterwards, global solutions are constructed by stitching local solutions. Finally, employing the $θ$-method, we explore the existence and the uniqueness of global solutions for diagonally quadratic mean-field BSDEs with convex generators, even in the case of unbounded terminal values that have exponential moments of all orders. These results are extended to a Volterra-type case where the coefficients can even be of quadratic growth with respect to the law of $Z$.