We define the Unbounded Inner-Product functionality in the context of Public-Key Functional Encryption, and introduce schemes that realize it under standard assumptions. In an Unbounded Inner-Product Functional Encryption scheme, a public key allows anyone to encrypt unbounded vectors, that are essentially mappings from N∗ to Zp. The owner of the master secret key can generate functional decryption keys for other unbounded vectors. These keys enable one to evaluate the inner product between the unbounded vector underlying the ciphertext and the unbounded vector in the functional decryption key, provided certain conditions on the two vectors are met. We build Unbounded Inner-Product Functional Encryption by introducing pairings, using a technique similar to that of Boneh-Franklin Identity-Based Encryption. A byproduct of this is that our scheme can be made Identity-Based "for free". It is also the first Public-Key Inner-Product Functional Encryption Scheme with a constant-size public key (and master secret key), as well constant-size functional decryption keys: each consisting of just one group element.