As an important branch of electromagnetic metamaterials, hyperbolic metamaterials become the focus of research for their unique property of controlling near-field waves. Hyperbolic metasurface is a new type of planar metamaterials with hyperbolic dispersion relationship and has many similarities in theory and applications with hyperbolic metamaterial. Compared with the bulk hyperbolic metamaterials, hyperbolic metasurfaces exhibit more excellent performances because the large reduction in the longitudinal dimension limits the propagation of the electromagnetic waves in the two-dimensional plane. In the first part of this review, we introduce hyperbolic metamaterial with its theory, implementation and applications. The latter part of the review is about hyperbolic metasurfaces and their potential applications. We also point out the restrictions of the hyperbolic metamaterials and metasurfaces and the prospect of future applications.