Editor's Note. Introduction. Chapter 1: The Inoariant Measure for Pla~ies in Space. Finite algebras and invariant imbedding. A bit of geometrical analysis. Wedges and Buffon rings. Chapter 2: Liw~es in the Plane and Other Families of Curves. Degeneration of wedges into I~eedles. Non-invariant measures. Topological invariance; discoids and G-oids. Diophantine representation in the whole sphere. Riemann and spherical cabbages. Chapter 3: Alternative Proofs in Two Dimer~ions. Sylvester rings and Sylvester's garland. Extension of Crofton's theorem. Topological avoroach and the Gauss-Bonnet theorem. Uniqueness of Diophantine decompositions. Chapter 4: Measures on Products of Line Spaces. Buffon sets in G ~'. Measures on the space of triangles. Non-random convex domains. Chapter 5: Dimensions Three and Your, and Beyond. Planes in space: from needles to wedges. Four dimensions: tethedra as tools for integration. Invariant imbeddings. Chapter 6: Geometrical Inequalities and the Construction o*. Measures. Pseudomettics in. the plane. From plane to discoid. Products of line spaces. Other spaces. Extension outwards. Chapter 7: PleijeI-Type Identities for Convex Domains. Pleijel identities for discoids. Integrals over r. Integrals over E. Chapter 8: Shadings of Convex Domains. The main idea. Geometrical integration. A geometrical inequality--and a check. Application to black/white shadings. One special shading. Chapter 9: An Application to Stereology of Polygon Processes. Preliminaries. Mathematical stereology. Associated marked point processes: side length distribution. Vertex angle distribution Mean perimeter length. Remarks on random polyhedron processes. Chapter 10: An Application to Stereology of Ergodic Boolean Models in R 2. Shadings of quadrilaterals. Summation over quadrilaterals. Ergodic random domain processes. Some integral geometry in the small. Justifi,.ation by dominated convergence in the case of restricted curvature. Black-reeurret~.ce and exponentiality. Conchtsion. The Buff'on-Sylvester problem. Enumeration of atoms, r-Dimensional planes of R". Inequalities Decompositions in general. Triangles. Essential inequalities. Random packing. ".rest interval.,. Classes of equivalent matrices. References. Appendix A: by Adrian Baddeley. lntro,at~ction. Combinatorial aspects of integral geometry. Combinatorial integral geometry. Mea:;ures on G. Construction of measures. Directed lines. Higher dimensions and non-Euclidean spaces. Measures and measurements. Hilbert's fourth problem. References for Appendix A. Appendix I5: Stereology of Polyhedra: Numerical results by R.V. Amba,tzumian Tetrahedra. Right triangular prisms. Rectangular parallelepipeds. Right hexagonal prisms. Index.