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The Tera Electronvolt Superconducting Linear Accelerator TESLA is the only linear electron-positron collider project based on superconductor technology for particle acceleration. In the first stage with 500 GeV center-of-mass energy an accelerating field of 23.4 MV/m is needed in the superconducting niobium cavities which are operated at a temperature of 2 K and a quality factor $Q_{0}$ of $10^{10}$. This performance has been reliably achieved in the cavities of the TESLA Test Facility (TTF) accelerator. The upgrade of TESLA to 800 GeV requires accelerating gradients of 35 MV/m. Using an improved cavity treatment by electrolytic polishing it has been possible to raise the gradient to 35 - 43 MV/m in single cell resonators. Here we report on the successful transfer of the electropolishing technique to multi-cell cavities. Presently four nine-cell cavities have achieved 35 MV/m at $Q_{0}$ ≥ $5$ × $10^{9}$, and a fifth cavity could be excited to 39 MV/m. In two high-power tests it could be verified that EP-cavities preserve their excellent performance after welding into the helium cryostat and assembly of the high-power coupler.One cavity has been operated for 1100 hours at the TESLA-800 gradient of 35 MV/m and 57 hours at 36 MV/m without loss in performance.

The 18th International Conference on High Energy Accelerators, HEACC2001, Tsukuba, Japan, 26 Mar 2001 - 30 Mar 2001; 4-8 (2001)., The cavity research and development which was essential to achieve the performance goals of a Q$_0$ of 10$^{10}$ at an accelerating gradient of 23,5 MV/m for TESLA-500 will be reviewed. Results from the 1,3 GHz nine-cell superconducting niobium cavities will be shown. For TESLA-800 the specifications are a Q$_0$=5×$10^9$ for the accelerating gradient of 35 MV/m. It will be shown that this is achieved regularly in electropolished one-cell cavities. First promising results on nine-cell cavities are shown. For TESLA-800 it is necessary to increase the fill factor of the linac. This is achieved via the superstructure concept. The first succesful test with a piezoelectric tuner to compensate the frequency detuning during the rf pulse is shown.