A parameter $c_0\in\mathbb{C}$ in the family of quadratic polynomials $f_c(z)=z^2+c$ is a critical point of a period $n$ multiplier, if the map $f_{c_0}$ has a periodic orbit of period $n$, whose multiplier, viewed as a locally analytic function of $c$, has a vanishing derivative at $c=c_0$. Information about the location of the critical points and critical values of the multipliers can be helpful for understanding the geometry of the Mandelbrot set. In this talk we will review some results about the critical points of the multipliers in the quadratic family and discuss some related results for other families of maps.