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Binary-state models are those in which the constituent elements can only appear in two possible configurations. These models are fundamental in the mathematical treatment of a number of phenomena such as spin interactions in magnetism, opinion dynamics, rumor and information spreading in social systems, etc. Here, we focus on the study of non-Markovian effects associated with aging for binary-state dynamics in complex networks. Aging is considered as the property of the agents to be less prone to change state the longer they have been in the current state, which gives rise to heterogeneous activity patterns. We analyze in this context the Threshold model of Complex Contagion, which has been proposed to explain, for instance, processes of adoption of new technologies and in which the agents need the reiterated confirmation of several contacts (until reaching over a given neighbor fraction threshold) to change state. Our analytical approximations give a good description of extensive numerical simulations in Erdös-Rényi, random-regular and Barabási-Albert networks. While aging does not modify the spreading condition, it slows down the cascade dynamics towards the full-adoption state: the exponential increase of adopters in time from the original model is replaced by a stretched exponential or power-law, depending on the aging mechanism. Under several approximations, we give analytical expressions for the cascade condition and for the exponents of the exponential, power-law and stretched exponential growth laws for the adopters density. Beyond networks, we also describe by numerical simulations the effects of aging for the Threshold model in a two-dimensional lattice.