The chapter introduces the concept of iterated path integrals on differentiable spaces, which are generalizations of line integrals. These integrals are constructed by iteratively integrating differential forms over paths in a manifold. The chapter covers the following key aspects: 1. **Definition and Properties of Iterated Integrals**: Iterated integrals are defined as repeated integrations of differential forms over paths. They are shown to be useful in relating analysis on manifolds to the homology of path spaces. 2. **Geometrical Significance**: Iterated integrals play a crucial role in connecting analysis and algebraic topology. For example, the real loop space cohomology of a simply connected compact manifold is isomorphic to the cohomology of the complex of iterated integrals on the smooth loop space. 3. **Computational Tools**: The de Rham theoretical approach provides computational tools for dealing with commutative differential graded algebras, which are useful in computing the homology and cohomology of loop spaces and other path spaces. 4. **Differentiable Spaces**: The chapter introduces the concept of differentiable spaces, which are sets equipped with a family of set maps called plots. These spaces generalize manifolds and are used to define iterated integrals and their properties. 5. **Integration and Chain Complexes**: The chapter discusses the integration of differential forms over paths and the construction of chain complexes from these integrals. It introduces the notion of plot chains and their homology, which is a fundamental tool in understanding the geometry of path spaces. 6. **Poincaré Operators**: The chapter defines Poincaré operators, which are used to relate iterated integrals to homotopy invariance. These operators are essential for proving theorems about the cohomology of path spaces. 7. **Products of Path Space Plots**: The chapter defines two partial multiplications of plots in the path space, which are used to further explore the geometrical significance of iterated integrals. These multiplications are shown to commute with integration. 8. **Loop Space Cohomology**: The chapter concludes with a discussion on the cohomology of loop spaces. It presents a theorem that characterizes the cohomology of the restriction of a differential graded subalgebra of the de Rham complex to the loop space. This theorem is particularly useful for understanding the cohomology of simply connected manifolds. Overall, the chapter provides a comprehensive introduction to the theory of iterated path integrals and their applications in differential geometry and topology.The chapter introduces the concept of iterated path integrals on differentiable spaces, which are generalizations of line integrals. These integrals are constructed by iteratively integrating differential forms over paths in a manifold. The chapter covers the following key aspects: 1. **Definition and Properties of Iterated Integrals**: Iterated integrals are defined as repeated integrations of differential forms over paths. They are shown to be useful in relating analysis on manifolds to the homology of path spaces. 2. **Geometrical Significance**: Iterated integrals play a crucial role in connecting analysis and algebraic topology. For example, the real loop space cohomology of a simply connected compact manifold is isomorphic to the cohomology of the complex of iterated integrals on the smooth loop space. 3. **Computational Tools**: The de Rham theoretical approach provides computational tools for dealing with commutative differential graded algebras, which are useful in computing the homology and cohomology of loop spaces and other path spaces. 4. **Differentiable Spaces**: The chapter introduces the concept of differentiable spaces, which are sets equipped with a family of set maps called plots. These spaces generalize manifolds and are used to define iterated integrals and their properties. 5. **Integration and Chain Complexes**: The chapter discusses the integration of differential forms over paths and the construction of chain complexes from these integrals. It introduces the notion of plot chains and their homology, which is a fundamental tool in understanding the geometry of path spaces. 6. **Poincaré Operators**: The chapter defines Poincaré operators, which are used to relate iterated integrals to homotopy invariance. These operators are essential for proving theorems about the cohomology of path spaces. 7. **Products of Path Space Plots**: The chapter defines two partial multiplications of plots in the path space, which are used to further explore the geometrical significance of iterated integrals. These multiplications are shown to commute with integration. 8. **Loop Space Cohomology**: The chapter concludes with a discussion on the cohomology of loop spaces. It presents a theorem that characterizes the cohomology of the restriction of a differential graded subalgebra of the de Rham complex to the loop space. This theorem is particularly useful for understanding the cohomology of simply connected manifolds. Overall, the chapter provides a comprehensive introduction to the theory of iterated path integrals and their applications in differential geometry and topology.