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We consider the problem of making apps fault-tolerant through replication, when apps operate at the microsecond scale, as in finance, embedded computing, and microservices apps. These apps need a replication scheme that also operates at the microsecond scale, otherwise replication becomes a burden. We propose Mu, a system that takes less than 1.3 microseconds to replicate a (small) request in memory, and less than a millisecond to fail-over the system - this cuts the replication and fail-over latencies of the prior systems by at least 61% and 90%. Mu implements bona fide state machine replication/consensus (SMR) with strong consistency for a generic app, but it really shines on microsecond apps, where even the smallest overhead is significant. To provide this performance, Mu introduces a new SMR protocol that carefully leverages RDMA. Roughly, in Mu a leader replicates a request by simply writing it directly to the log of other replicas using RDMA, without any additional communication. Doing so, however, introduces the challenge of handling concurrent leaders, changing leaders, garbage collecting the logs, and more - challenges that we address in this paper through a judicious combination of RDMA permissions and distributed algorithmic design. We implemented Mu and used it to replicate several systems: a financial exchange app called Liquibook, Redis, Memcached, and HERD. Our evaluation shows that Mu incurs a small replication latency, in some cases being the only viable replication system that incurs an acceptable overhead.