We survey recent work on local well-posedness results for parabolic equations and systems with rough initial data. The design of the function spaces is guided by tools and constructions from harmonic analysis, like maximal functions, square functions and Carleson measures. We construct solutions under virtually optimal scale invariant conditions on the initial data. Applications include BMO initial data for the harmonic map heat flow and the Ricci-DeTurck flow for initial metrics with small local oscillation. The approach is sufficiently flexible to apply to boundary value problems, quasilinear and fully nonlinear equations.