n-DBI gravity is a gravitational theory motivated by Dirac-Born-Infeld type conformal scalar theory and designed to yield non-eternal inflation spontaneously without introducing any scalar field. It contains the space-time foliation provided by an everywhere time-like vector field n, which couples to the gravitational sector of the theory, but decouples in the small curvature limit. In my talk, I will explain some theoretical aspects of the model in detail by showing its interesting black hole solutions.