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One approach to solving the problem of quantum gravity is based on the causal set hypotheis, which states that the deep, quantum structure of spacetime is discrete and is what is known in mathematics as a “partial order” or “poset”, a kind of extended family tree. Causal set theory has now reached a stage at which questions of phenomenology are beginning to be addressed. this talk will introduce the basic concepts and motivations behind the hypothesis and address some of the latest developments which include: (i) an apparently confirmed order of magnitude prediction for the cosmological constant, the only prediction made in any propsed theory of quantum gravity that has been subsequently verified by observation (ii) a classical stochastic causal set dynamics which arguably is the most general consistent with the discrete analogs of general covariance and classical casuality (iii) a rigorous characterization of the observables (or “physical quetions”) of causal set cosmology,at least in the classical case. (iv) a model of detectable phenomenological effects of spacetime discreteness.