Behaviours of light in materials strongly depend on the topological structure of the iso-frequency surface (IFS). The usual materials, of which the unit cell of photonic crystal is made up, are dielectrics, whose IFSs have the same closed topological structure. As a simplest photonic crystal, one-dimensional photonic crystal (1DPC) has attracted intensive attention due to its simple fabrication technique as well as numerous applications. However, in a conventional all-dielectric 1DPC, photonic band gaps (PBGs) for both transverse magnetic (TM) and transverse electric (TE) polarizations will shift toward short wavelengths (i.e. blueshift) as incident angle increases. The underlying physical reason is that the propagating phase in isotropic dielectric will decrease as incident angle increases. The blueshift property of band gap for TM and TE polarization will limit the band width of omnidirectional band gap and the range of operating incident angles in some PBG-based applications, including near-perfect absorption, polarization selection and sensitive refractive index sensing. However, for TM polarization, the propagating phase in a hyperbolic metamaterial (HMM) will increase with incident angle increasing. This special phase property of HMM provides us with a way to flexibly tune the angle-dependent property of band gap in periodic compound structure composed of alternative HMM with open IFS and dielectric with close IFS. In this review, we realize zeroshift (i.e. angle-independent) band gaps as well as redshift band gaps in 1DPCs containing HMMs, which can be utilized to realize near-perfect absorption, sensitive refractive index sensing and polarization selection working in a wide range of incident angles.