  The purpose of this paper is to investigate general mean-field backward
stochastic differential equations (MFBSDEs) in multi-dimension with diagonally
quadratic generators $f(\omega,t,y,z,\mu)$, 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 $\theta$-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 $Z$.
