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A natural approach to study triangulated manifolds is to try and construct them 'one piece at the time'. Variations of this idea have been studied in several fields of mathematics, from convex polytopes ('shellings') to combinatorial commutative algebra ('constructions'), from discrete quantum gravity ('local constructions') to differential topology ('handle decompositions'). We compare these approaches and discuss connections with (discrete) Morse theory. If time permits, we present a few combinatorial applications.