The principal aim of the paper is to discuss and revise the derivation of different realizations of the Cauchy kernel \(K(\lambda, \mu)={1 \over P(\lambda)-P (\mu)}\), on Riemann surfaces, where \(\lambda\) and \(\mu\) are local coordinates of points \(P(\lambda)\), \(P(\mu)\). The first realization leads to the expression \(K(\lambda,\mu) =\sum^\infty_1 {d\Omega_j\over d \Omega_1} (\lambda) {\mu^{-1} \over j}\), where \(\partial\Omega_s\) is normalized second kind meromorphic differential with principal part of the unique pole of the form \(-sz^{-s-1}dx\). This expression is shown to be written in terms of the local Cauchy kernel \(Z_p(x)= (d/dx) \log E(x,p)\), where \(E(x,p)\) is a Fay-Klein prime form, which is given in terms of \(\theta\)-functions and \(-1/2\)-holomorphic differentials. The second realization is given in terms of the Baker-Akhiezer function of the curve and then the Cauchy-Baker-Akhiezer kernel is given as \[ d\omega (k,k',x,y,t, \dots)= {1\over 2 \pi i} \int^x_{\pm \infty} \psi(k, x',y,t, \dots) \psi^† (k',x',y,t, \dots) dx', \] where \(x,y,t, \dots= {\mathbf t}\) are the “times” of integrable hierarchy of KP-type, the Baker-Akhiezer function \(\psi\) is expanded near the essential singularity as \(\exp [\xi({\mathbf t}, k)] \cdot(1+ \sum^\infty_1 \chi_i k^{-i})\) where \(k\) is a local coordinate, the function \(\psi^†= \psi^* d\widehat \Omega\sim \exp[\xi( {\mathbf t},k)] \cdot (1+\sum^\infty_1 \chi^*_i k^{-i}(1+ \beta/k^2+ \dots))\), where \(d\widehat \Omega\) is the meromorphic differential with zeros in \(D+D^*\) and double pole at \(\infty\). The kernel construction developed in the paper was applied to solve the dispersionless KP hierarchy and show its equivalency to the dispersionless differential Fay identity. The paper contains a preliminary material on Riemann surfaces. Baker-Akhiezer functions, integrable hierarchies, Hirota bilinear equations being based on the previous publications of the author. In particular the realization of \(\tau\) functions of the dispersionless KP in terms of Schur polynomials was given.