We will consider a class of Calabi-Yau varieties given by cyclic branched covers of a fixed semi Fano manifold. The first prototype example goes back to Euler, Gauss and Legendre, who considered 2-fold covers of P1 branched over 4 points. Two-fold covers of P2 branched over 6 lines have been studied more recently by many authors, including Matsumoto, Sasaki, Yoshida and others, mainly from the viewpoint of their moduli space and their comparisons. I will outline a higher dimensional? ?generalization from the viewpoint of mirror symmetry. We will introduce a new compactication of the moduli space cyclic covers, using the idea of `abelian gauge fixing' and `fractional complete intersections'. This produces a moduli problem that is amenable to tools in toric geometry, particularly those that we have developed jointly in the mid-90's with S. Hosono and S.-T. Yau in our study of toric Calabi-Yau complete intersections. In dimension 2, this construction gives rise to new and interesting identities of modular forms and mirror maps associated to certain K3 surfaces. We also present an essentially complete mirror theory in dimension 3, and discuss generalization to higher dimensions. The lecture is based on on-going joint work with S. Hosono, T.-J. Lee, H. Takagi, S.-T. Yau.