Holographic quantum field theories that confine in flat space, are considered on a fixed AdS space. The space of holographic solutions for such theories is constructed and three types of regular solutions are found. Theories with two AdS boundaries provide interfaces between two confining theories. Theories with a single AdS boundary correspond to ground states of a single confining theory on AdS. We find solutions without a boundary, whose interpretation is not obvious. There is also a special limiting solution that oscillates an infinite number of times around the UV fixed point. We analyze in detail the holographic dictionary for the one-boundary solutions and compute the free energy. No (quantum) phase transitions are found when we change the curvature. We find an infinite number of pure vev solutions, but no CFT solution without a vev. We also compute the free energy of the interface solutions. We find that the product saddle points have always lower free energy than the connected solutions. This implies that in such interfaces, normalized cross-correlators vanish exponentially in N_c^2.

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