  The conceptional design of the proposed linear electron-positron collider
TESLA is based on 9-cell 1.3 GHz superconducting niobium cavities with an
accelerating gradient of Eacc >= 25 MV/m at a quality factor Q0 > 5E+9. The
design goal for the cavities of the TESLA Test Facility (TTF) linac was set to
the more moderate value of Eacc >= 15 MV/m. In a first series of 27
industrially produced TTF cavities the average gradient at Q0 = 5E+9 was
measured to be 20.1 +- 6.2 MV/m, excluding a few cavities suffering from
serious fabrication or material defects. In the second production of 24 TTF
cavities additional quality control measures were introduced, in particular an
eddy-current scan to eliminate niobium sheets with foreign material inclusions
and stringent prescriptions for carrying out the electron-beam welds. The
average gradient of these cavities at Q0 = 5E+9 amounts to 25.0 +- 3.2 MV/m
with the exception of one cavity suffering from a weld defect. Hence only a
moderate improvement in production and preparation techniques will be needed to
meet the ambitious TESLA goal with an adequate safety margin. In this paper we
present a detailed description of the design, fabrication and preparation of
the TESLA Test Facility cavities and their associated components and report on
cavity performance in test cryostats and with electron beam in the TTF linac.
The ongoing R&D towards higher gradients is briefly addressed.
