The idea of linearising functions locally has been giving birth to differential calculus by Leibniz and Newton. Three centuries later, this idea is still important, for example in nonlinear parabolic pde's.
To solve these (at least locally in time), fixed point arguments can be used, and to set up these, the notion of $L_p$ maximal regularity arises. In the most easy setting, it is the question whether the solution $x$ to the Cauchy problem \[x'(t) + A x(t) = f(t)\,,\qquad     x(0)=0\] satisfies $x' \in L_p$ and $A x \in L_p$ whenever $f \in L_p$. We will overview the situation first for autonomous equations in Hilbert spaces, then pass to UMD Banach spaces and then shift to non-autonomous equations.
This survey talk aims to sketch the general lines as well as  give some
insight in key ideas (but not technicalities) of the proofs.