Public-Key Encryption (PKE) and Message Authentication (PKMA, aka as digital signatures) are fundamental cryptographic primitives. Traditionally, both notions are defined as non-interactive (i.e., single-message). In this work, we initiate rigorous study of (possibly) interactive PKE and PKMA schemes. We obtain the following results demonstrating the power of interaction to resolve questions which are either open or impossible in the non-interactive setting. Efficiency/Assumptions. One of the most well known open questions in the area of PKE is to build, in a “black-box way”, so called chosen ciphertext attack (CCA-) secure PKE from chosen plaintext attack (CPA-) secure PKE. In contrast, we show a simple 2-round CCA-secure PKE from any (non-interactive) CPA-secure PKE (in fact, these primitives turn out to be equivalent). Similarly, although non-interactive PKMA schemes can be inefficiently built from any one-way function, no efficient signature schemes are known from many popular numbertheoretic assumptions, such as factoring, CDH or DDH. In contrast, we show an efficient 2-round PKMA from most popular assumptions, including factoring, CDH and DDH. Advanced Properties. It is well known that no non-interactive signature (resp. encryption) scheme can be deniable (resp. forward-secure), since the signature (resp. ciphertext) can later “serve as an evidence of the sender’s consent” (resp. “be decrypted if the receiver’s key is compromised”). We also formalize a related notion of replay-secure (necessarily) interactive PKMA (resp. PKE) schemes, where the verifier (resp. encryptor) is assured that the “current” message can only be authenticated (resp. decrypted) by the secret key owner now, as opposed to some time in the past (resp. future). We observe that our 2-round PKMA scheme is both replay-secure and (passively) deniable, and our 2-round PKE scheme is both replay- and forward-secure.