XTE J1739–302 is a transient X-ray source with unusually short outbursts, lasting on the order of hours. Here we give a summary of X-ray observations we have made of this object in outburst with the Rossi X-ray Timing Explorer (RXTE) and at a low level of activity with the Chandra X-ray Observatory, as well as observations made by other groups. Visible and infrared spectroscopy of the mass donor of XTE J1739–302 are presented in a companion paper. The X-ray spectrum is hard both at low levels and in outburst, but somewhat variable, and there is strong variability in the absorption column from one outburst to another. Although no pulsation has been observed, the outburst data from multiple observatories show a characteristic timescale for variability on the order of 1500–2000 s. The Chandra localization (right ascension 17 h 39 m 11.58 s, declination −30 ◦ 20 ′ 37.6 ′ ′ , J2000) shows that despite being located less than 2 ◦ from the Galactic Center and highly absorbed, XTE J1739–302 is actually a foreground object with a bright optical counterpart. The combination of a very short outburst timescale and a supergiant companion is shared with several other recently-discovered systems, forming a class we designate as Supergiant Fast X-ray Transients (SFXTs). Three persistently bright X-ray binaries with similar supergiant companions have also produced extremely short, bright outbursts: Cyg X–1, Vela X–1, an
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