  In this paper we consider a class of stochastic reaction-diffusion equations.
We provide local well-posedness, regularity, blow-up criteria and positivity of
solutions. The key novelties of this work are related to the use transport
noise, critical spaces and the proof of higher order regularity of solutions --
even in case of non-smooth initial data. Crucial tools are $L^p(L^q)$-theory,
maximal regularity estimates and sharp blow-up criteria. We view the results of
this paper as a general toolbox for establishing global well-posedness for a
large class of reaction-diffusion systems of practical interest, of which many
are completely open. In our follow-up work (Agresti and Veraar: Global
existence ... ), the results of this paper are applied in the specific cases of
the Lotka-Volterra equations and the Brusselator model.
