In an Instance-Hiding Interactive Proof (IHIP) [Beaver et al. CRYPTO 90], an efficient verifier with a _private_ input x interacts with an unbounded prover to determine whether x is contained in a language L. In addition to completeness and soundness, the instance-hiding property requires that the prover should not learn anything about x in the course of the interaction. Such proof systems capture natural privacy properties, and may be seen as a generalization of the influential concept of Randomized Encodings [Ishai et al. FOCS 00, Applebaum et al. FOCS 04, Agrawal et al. ICALP 15], and as a counterpart to Zero-Knowledge proofs [Goldwasser et al. STOC 89].

We investigate the properties and power of such instance-hiding proofs, and show the following:
1. Any language with an IHIP is contained in AM/poly and coAM/poly.
2. If an average-case hard language has an IHIP, then One-Way Functions exist.
3. There is an oracle with respect to which there is a language that has an IHIP but not an SZK proof.
4. IHIP's are closed under composition with any efficiently computable function.

We further study a stronger version of IHIP (that we call Strong IHIP) where the view of the honest prover can be efficiently simulated. For these, we obtain stronger versions of some of the above:
5. Any language with a Strong IHIP is contained in AM and coAM.
6. If a _worst-case_ hard language has a Strong IHIP, then One-Way Functions exist.