To solve generalized equations of type \(0 \in f(x) + F(x)\) with \(f:X\rightarrow \mathbb R, F:X\Rightarrow Y\) , \(X,Y\) Banach spaces, \(f\) a continuous function, \(F\) a set-valued function with closed graph, \(f\) and \(F\) possibly nonsmooth, new Newton methods are given under convergence conditions of Kantorovich type, which means, that such conditions are related to the starting point (in some cases also during the iteration process). The first five pages (Introduction) deal with historical remarks on Newton methods. Especially there are a version of Kantorovich's theorem (as in his book with Akilov), a modified statement of \textit{R. G. Bartle}'s theorem (originally in [Proc. Am. Math. Soc. 6, 827--831 (1955; Zbl 0066.09805)]) and finally remarks about (as the authors write) a path-breaking paper by \textit{L. Qi} and \textit{J. Sun} [Math. Program. 58, No. 3 (A), 353--367 (1993; Zbl 0780.90090)], where Clarke's generalized Jacobian \(\partial f\) is used (shortly or more generally \(A_k\) in step \(k\), \(A_k\) a linear bounded operator). A convergence theorem from the paper by Qi and Sun is given (with \(f\)) and then, in Section 2, a strengthened version can be found with full proof followed (now with \(f\) and \(F\) and metric regularity for set-valued mappings as one of the conditions) by the main theorem of the paper. This new theorem contains several results on convergence rates. The last three sections give different special cases, examples, (also explicit) numerical results and comments, showing the experience of the authors in this field.