What nonlinear extension of ARMA and State Space Model do exist?

In ARMA model we postulate that predictions of time series can be calculated as a linear function of $N$ previous observations (AR part) and $K$ differences between the previous observations and the corresponding predictions (MA part).

Are there extensions of this linear model in which we use a nonlinear function of previous observations and predictions?

Are there even more general models in which we also calculate some auxiliary parameters as nonlinear functions of previous observations, predictions and auxiliary parameters and then use these auxiliary parameters to calculate next predictions (and next values of the auxiliary parameters)?

If got it right, the sate space model goes in approximately this direction by introducing auxiliary (or hidden) parameters but in this model everything is linear. Are there nonlinear extensions of the state space model?

Can somebody explain how Gaussianity of noise is related to linearity of the model. Why we cannot have a nonlinear function of previous observations as an estimation of mean and then have a Gaussian noise on top of that. For example: $y_i = \alpha \cdot y_{i-1}^2 + \beta \cdot y_{i-2}^3 + n(0, \sigma^2)$