The silicon vertex tracker (SVT) of the BaBar experiment at PEP-II is described. This is the crucial device for the measurement of the B meson decay vertices to extract charge-conjugation parity (CP) asymmetries. It consists of five layers of double-sided ac-coupled silicon strip detectors, read out by a full-custom integrated circuit, capable of simultaneous acquisition, digitization, and transmission of data. It represents the core of the BaBar tracking system, providing position measurements with a precision of 10 μm (inner layers) and 30 μm (outer layers). The relevant performances of the SVT are presented, and the experience acquired during the construction, installation, and the first five years of data-taking is described. Innovative solutions are highlighted, like the sophisticated alignment procedure, imposed by the design of the silicon tracker, integrated in the beamline elements and mechanically separated from the other parts of BaBar. The harshness of the background conditions in the interaction region required several studies on the radiation damage of the sensors and the front-end chips, whose results are presented. Over the next five years the luminosity is predicted to increase by a factor three, leading to radiation and occupancy levels significantly exceeding the detector design. Extrapolation of future radiation doses and occupancies is shown together with the expected detector performance and lifetime. Upgrade scenarios to deal with the increased luminosity and backgrounds are discussed.