This paper presents a new geometric and physical interpretation of fractional integration and differentiation. The author discusses the geometric interpretation of left- and right-sided Riemann–Liouville fractional integrals, the Riesz potential, and the Feller potential. The geometric interpretation is based on the idea of "shadows" cast by a "fence" on two walls, representing the integral and fractional integral. The physical interpretation of the Riemann–Liouville fractional integration is proposed in terms of inhomogeneous and changing time scales. The author also provides a new physical interpretation of the Stieltjes integral. The paper also discusses the physical interpretation of the Riemann–Liouville and Caputo fractional derivatives, and shows that the suggested geometric interpretation can be used for providing a new geometric and physical interpretation for convolution integrals of the Volterra type. The author argues that the physical interpretation of fractional integration is in line with current views on space-time in physics. The paper also discusses the concept of two kinds of time: cosmic time and individual time, and how they relate to the physical interpretation of fractional integration. The author concludes that the geometric and physical interpretation of fractional integration and differentiation provides a new understanding of these operations and their applications in various fields.This paper presents a new geometric and physical interpretation of fractional integration and differentiation. The author discusses the geometric interpretation of left- and right-sided Riemann–Liouville fractional integrals, the Riesz potential, and the Feller potential. The geometric interpretation is based on the idea of "shadows" cast by a "fence" on two walls, representing the integral and fractional integral. The physical interpretation of the Riemann–Liouville fractional integration is proposed in terms of inhomogeneous and changing time scales. The author also provides a new physical interpretation of the Stieltjes integral. The paper also discusses the physical interpretation of the Riemann–Liouville and Caputo fractional derivatives, and shows that the suggested geometric interpretation can be used for providing a new geometric and physical interpretation for convolution integrals of the Volterra type. The author argues that the physical interpretation of fractional integration is in line with current views on space-time in physics. The paper also discusses the concept of two kinds of time: cosmic time and individual time, and how they relate to the physical interpretation of fractional integration. The author concludes that the geometric and physical interpretation of fractional integration and differentiation provides a new understanding of these operations and their applications in various fields.