We present the first good evidence for exocomet transits of a host star in continuum light in data from the Kepler mission. The Kepler star in question, KIC 3542116, is of spectral type F2V and is quite bright at Kp = 10. The transits have a distinct asymmetric shape with a steeper ingress and slower egress that can be ascribed to objects with a trailing dust tail passing over the stellar disc. There are three deeper transits with depths of ≃ 0.1 per cent that last for about a day, and three that are several times more shallow and of shorter duration. The transits were found via an exhaustive visual search of the entire Kepler photometric data set, which we describe in some detail. We review the methods we use to validate the Kepler data showing the comet transits, and rule out instrumental artefacts as sources of the signals. We fit the transits with a simple dust-tail model, and find that a transverse comet speed of ˜35-50 km s-1 and a minimum amount of dust present in the tail of ˜1016 g are required to explain the larger transits. For a dust replenishment time of ˜10 d, and a comet lifetime of only ˜300 d, this implies a total cometary mass of ≳3 × 1017 g, or about the mass of Halley's comet. We also discuss the number of comets and orbital geometry that would be necessary to explain the six transits detected over the 4 yr of Kepler prime-field observations. Finally, we also report the discovery of a single comet-shaped transit in KIC 11084727 with very similar transit and host-star properties.
