This is a very important book whose subject matter ranges much more widely than the title suggests. Chapter 0 gives a brief history of the problem, which we now summarise as it gives a good indication of the purpose of the book-essentially an attempted classification of a special class of Riemannian manifolds. More specifically, consider the two assertions:SC: there exists a positive real number I such that for every vector £ in the unit tangent bundle UM of (M, g), the geodesic y with initial velocity vector y(0) = £ is (i) a periodic map with least period /,SC m : there exists a point m in M such that the above property is supposed to hold for every £ in the unit sphere U m at m (i.e. the assumption holds on m only).The whole book is devoted to what may appear to be the narrow question of classifying SC Riemannian manifolds to within isometry and SC" manifolds to within diffeomorphism. It is not difficult to prove that compact symmetric spaces of rank one, namely (S d , can), (RF 1 , can), (CP d , can), (HP d , can), (CaP 2 , can), with their canonical Riemannian structure are examples of SC manifolds.The first attempt to consider non-canonical metrics on S 2 satisfying condition SC appears to go back to Darboux, but he did not establish the existence of such a metric globally. After preliminary attempts by Tannery at the end of the 19th century, the question was settled by Zoll in 1903 who constructed a real analytic Riemannian manifold of revolution (S 2 , g) with the SC property.Natural questions arise such as "Are there many non-isometric SC Riemannian manifolds (S 2 , #)?". "Are there such SC Riemannian structures on the other candidate manifolds, RP d , S d (d ^ 3), CP 6 (d ^ 2), HP 4 (d > 2), CaP 2 ?". Little is known about this problem in general, apart from the case of S d -and even here results are quite recent. The only general result known is due to A. Weinstein (1974). This states that in order that a d-dimensional Riemannian manifold (M,g) satisfy the SC property with / = 2n, it is necessary that its total volume shall be an integral multiple of the volume of (