Protected boundary modes originating from the topological non-triviality in the bulk are one of the central propositions of the topological band theory, which is termed the bulk-boundary correspondence. Topological photonic structures, governed by the electromagnetic-field generalization of topological band theory, have attracted significant attention due to their advantages as synthetic materials and promising applications of robust boundary modes in optical devices. However, due to the absence of Kramers' degeneracy in bosons and weak magnetic responses at optical frequencies, the realizations of topological phases in photonics are primarily based on crystalline symmetries, where the generalization of codimension-1 bulk-boundary correspondence might not exist. Therefore, extra caution is necessary when addressing the exact bulk-boundary correspondence in these photonic structures. Furthermore, phenomena originating from the non-trivial bulk topology in higher dimensional systems remain to be discovered. One representative example is topological flat bands which have a profound connection to the quantum geometry leading to the unconventional phenomena in correlated materials. These scenarios inspire an extension of these ideas to topological photonics, asking for efficient approaches to examine, characterize, and design topological flat bands. The recently developed symmetry-based method has become an ideal approach to diagnose and design topological structures. This thesis mainly covers analyses of the topological nature of representative photonic structures in 1D and 2D, focusing on exploring the connection between boundary modes and bulk topology based on its crystalline symmetries. The experimental characterization of coupled topological boundary modes in a 1D Su-Schrieffer-Heeger waveguide array is first discussed. We subsequently proceed to a 2D case where spin-momentum-locked boundary modes in a breathing honeycomb lattice are observed. In these experimental realizations, strict topological protections are compromised yet the boundary modes are considered to exhibit topological features, motivating further investigation of the exact bulk-boundary correspondence. Consequently, we re-examine the breathing honeycomb lattice and find the actual physical consequence of its topological properties and the cause of the featured boundary modes via crystalline symmetry-based indicators. Our work clarifies the understanding of the bulk topology and the origin of the observed boundary modes and can be extended to study other physical consequences of non-trivial bulk topology beyond boundary modes. We also present our initial proposal of a photonic realization of topological flat-bands, which may open new avenues for designing topological photonic systems.