This article provides a detailed insight into computational approaches for deciphering Bronze Age Aegean and Cypriot scripts, namely, the Archanes script and the Archanes formula, Phaistos Disk, Cretan hieroglyphic (including the Malia Altar Stone and Arkalochori Axe), Linear A, Linear B, Cypro-Minoan, and Cypriot scripts. The unique contributions of this article are threefold: (1) a thorough review of major Bronze Age Aegean and Cypriot scripts and inscriptions, digital data and corpora associated with them, existing computational decipherment methods developed in order to decipher them, and possible links to other scripts and languages; (2) the definition of 15 major challenges that can be encountered in computational decipherments of ancient scripts; and (3) an outline of a computational model that could possibly be used to simulate traditional decipherment processes of ancient scripts based on palaeography and epigraphy. In the context of this article the term decipherment denotes the process of discovery of the language and/or the set of symbols behind an unknown script, and the meaning behind it.