In this talk, contributions by Abel and Sonin to the origins of the general fractional integrals and derivatives with the Sonin kernels are first discussed. While Abel introduced and employed the integral and integro-differential operators with the power law kernels that are nowadays referred to as the Riemann–Liouville fractional integral and the Caputo fractional derivative, Sonin extended his method to the case of pairs of arbitrary kernels whose Laplace convolutions are identically equal to one.

Following this, we mention some recent results regarding the properties of general fractional integrals and derivatives with the Sonin kernels. In particular, the first and the second fundamental theorems of Fractional Calculus for the general fractional derivatives, the regularized general fractional derivatives, and the sequential general fractional derivatives are formulated in the appropriate spaces of functions.

As an application of this theory, we discuss the convolution series that are a far-reaching generalization of the power law series as well as a representation of functions in form of the generalized convolution Taylor series. Another application is the generalized convolution Taylor formula that contains convolution polynomials and remainders given in terms of the general fractional integrals and the general fractional sequential derivatives. A known case of this formula is the fractional Taylor formula with polynomials involving the power law functions with the fractional exponents and a remainder in terms of the Riemann–Liouville fractional integrals and the sequential Riemann–Liouville fractional derivatives.