Signcryption schemes provide an efficient messaging system for data that needs to be sent with data confidentiality, data integrity and data origin authentication. However, the bandwidth overhead for the use of signcryption in a network in which a large number of messages need to be sent may be high. Motivated by aggregate signature schemes, the authors propose the concept of an aggregate signcryption scheme. An aggregate signcryption scheme allows distinct signcryption ciphertexts intended for the same recipient to be merged into a single signcryption ciphertext of smaller size without losing any of their security guarantees. This has the potential to provide significant bandwidth savings.