Standard a posteriori Reduced-Order Models (ROM), namely generic Reduced Basis (RB) and specific Proper Orthogonal Decomposition (POD), are limited to a linear description of the reduced-dimensional manifold where the solution is sought. In other words, the solution space is bound to be Euclidean. However, the typical manifold containing all possible solutions of a complex parametric problem is far of being linear. Generally, a linear combination of two (or more) solutions does not necessarily correspond to another solution, associated with some other parametric values. Consequently, building the ROM strategy upon in a linear subspace is not optimal. This is because the dimension of this linear subspace (number of degrees of freedom of the reduced model) is often much larger than the actual dimension of the manifold described by the set of all solutions. Typically, the intrinsic dimension of the problem coincides with the number of independent parameters of the model. A linear dimensionality reduction strategy is unable to discover the intrinsic dimension. At the best, it guesses the lower Euclidean dimension including the actual nonlinear manifold. And this may largely overestimate the intrinsic dimension. The kernel POD (kPOD) idea is based on a kernel-driven nonlinear dimensionality reduction technique in order to discover the actual underlying dimension of the problem. Moreover, a proper definition of the kernel based on physical considerations (accounting for the prior physical knowledge) is proposed in order to obtain an optimal reduction. The main difficulty of these techniques lies in the backward mapping (from the reduced-dimension feature space to the original full-order description). Here, the backward mapping is built upon a local approximation (based on the neighbouring snapshots) complemented by a number of artificially generated snapshots, enriching the local approximation space. This is beyond the reach of a classical local Euclidean expansion. Again, the artificial snapshots are to be generated based on physical considerations, incorporating previous expertise.