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In this paper we develop a new approach to nonlinear stochastic partialdifferential equations with Gaussian noise. Our aim is to provide an abstractframework which is applicable to a large class of SPDEs and includes manyimportant cases of nonlinear parabolic problems which are of quasi- orsemilinear type. This first part is on local existence and well-posedness. Asecond part in preparation is on blow-up criteria and regularization. Ourtheory is formulated in an $L^p$-setting, and because of this we can deal withnonlinearities in a very efficient way. Applications to several concreteproblems and their quasilinear variants are given. This includes Burger'sequation, the Allen-Cahn equation, the Cahn-Hilliard equation,reaction-diffusion equations, and the porous media equation. The interplay ofthe nonlinearities and the critical spaces of initial data leads to new resultsand insights for these SPDEs. The proofs are based on recent developments inmaximal regularity theory for the linearized problem for deterministic andstochastic evolution equations. In particular, our theory can be seen as astochastic version of the theory of critical spaces due toPr\"uss-Simonett-Wilke (2018). Sharp weighted time-regularity allow us to dealwith rough initial values and obtain instantaneous regularization results. Theabstract well-posedness results are obtained by a combination of severalsophisticated splitting and truncation arguments.

Nonlinear parabolic stochastic evolution equations in critical spaces Part I. Stochastic maximal regularity and local existence