Motivated by privacy issues associated with dissemination of signed digital certificates, we define a new type of signature scheme called a lsquoUniversal Designated-Verifier Signaturersquo (UDVS). A UDVS scheme can function as a standard publicly-verifiable digital signature but has additional functionality which allows any holder of a signature (not necessarily the signer) to designate the signature to any desired designated-verifier (using the verifierrsquos public key). Given the designated-signature, the designated-verifier can verify that the message was signed by the signer, but is unable to convince anyone else of this fact. We propose an efficient deterministic UDVS scheme constructed using any bilinear group-pair. Our UDVS scheme functions as a standard Boneh-Lynn-Shacham (BLS) signature when no verifier-designation is performed, and is therefore compatible with the key-generation, signing and verifying algorithms of the BLS scheme. We prove that our UDVS scheme is secure in the sense of our unforgeability and privacy notions for UDVS schemes, under the Bilinear Diffie-Hellman (BDH) assumption for the underlying group-pair, in the random-oracle model. We also demonstrate a general constructive equivalence between a class of unforgeable and unconditionally-private UDVS schemes having unique signatures (which includes the deterministic UDVS schemes) and a class of ID-Based Encryption (IBE) schemes which contains the Boneh-Franklin IBE scheme but not the Cocks IBE scheme.