Order-Preserving Encryption (OPE) and leakage I recently came across OPE and Order-Revealing Encryption (ORE), but I didn't exactly got the idea why they are important. What's exactly important about them? Are they actually used in practice? My recent interest is leakage-resilient cryptography, and I saw there are some papers on leakage in OPE and ORE, but why is leakage important in OPE and ORE?
I do work in this area. OPE and ORE are important primarily because of their tremendous utility in building systems which can perform some computation on encrypted data. Contrary to general-purpose solutions like fully-homomorphic encryption, OPE and ORE can be used to provide drop-in (with no code change) security in applications like databases. They can do this with only a small decrease in efficiency of the application. 

OPE and ORE are used in practice. Companies like CipherCloud and Skyhigh Networks deploy encryption software for cloud applications like Salesforce that use OPE heavily. Other major tech companies are also looking at OPE and ORE, though I can't say which ones for confidentiality reasons.

The 'but' (in cryptography there's always a 'but') is that all practical OPE and ORE schemes leak some information about the plaintext. This may seem obvious - to have order-preserving encryption you kind of need to, you know, preserve order. The hitch is that most actual deployable OPE and ORE schemes leak more than just the order of the plaintexts, and in some sense this is unavoidable because of a well-known impossibility result by Boldyreva et al. Further, nobody in the research community really knows how bad this leakage even is for real applications of OPE and ORE - we have precise simulation-based proofs which bound the leakage, but these proofs assume uniformly-distributed plaintexts. It seems like most use cases of OPE and ORE do not involve uniformly-distributed values, though. Extending and generalizing these analyses is a major (and important) open question right now.

I should note that when research papers talk about the "leakage" of OPE, it's not quite the same "leakage" discussed in leakage-resilient cryptography (LRC). LRC is attempting to provide a formal framework for reasoning about the effect of side-channel attacks which leak bits of the secret key, but in analyses of OPE and ORE the key is assumed to be fully hidden from the adversary.