<p>The period geometry of Calabi-Yau <i>n</i>-folds — characterised by their variations of Hodge structure governed by Griffiths transversality, a graded Frobenius algebra, an integral monodromy and an intriguing arithmetic structure — is analysed for applications in string compactifications and to Feynman integrals. In particular, we consider type IIB flux compactifications on Calabi-Yau three-folds and elliptically fibred four-folds. After constructing suitable three-parameter three-folds, we examine the relation between symmetries of their moduli spaces and flux configurations. Although the fixed point loci of these symmetries are projective special Kähler, we show that a simultaneous stabilisation of multiple moduli on the intersection of these loci need not be guaranteed without the existence of symmetries between them. We furthermore consider F-theory vacua along conifolds and use mirror symmetry to perform a complete analysis of the two-parameter moduli space of an elliptic Calabi-Yau four-fold fibred over ℙ³. We use the relation between Calabi-Yau period geometries in various dimensions and, in particular, the fact that the antisymmetric products of one-parameter Calabi-Yau three-fold operators yield four-fold operators to establish pairs of flux vacua on the moduli spaces of the three- and four-fold compactifications. We give a splitting of the period matrix into a semisimple and nilpotent part by utilising the Frobenius structure. This helps bringing <i>ϵ</i>-dimensional regulated integration by parts relations between Feynman integrals into <i>ϵ</i>-factorised form and solve them by iterated integrals of the periods.</p>