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A service choreography defines a set of permitted sequences of message events as a specification for the interaction of services. Realizability is a fundamental sanity check for choreographies comparable to the notion of soundness for workflows. We study several notions of realizability: partial, distributed, and complete realizability. They establish increasingly strict conditions on realizing services. We investigate decidability issues under the synchronous and asynchronous communication models. For partial realizability, we show undecidability whereas the other two problems are decidable with reasonable complexity.