Miniaturized integrated optical coherence tomography (OCT) systems have the potential to unlock a wide range of both medical and industrial applications. This applies in particular to multi-channel OCT schemes, where scalability and low cost per channel are important, to endoscopic implementations with stringent size demands, and to mechanically robust units for industrial applications. We demonstrate that fully integrated OCT systems can be realized using the state-of-the-art silicon photonic device portfolio. We present two different implementations integrated on a silicon-on-insulator (SOI) photonic chip, one with an integrated reference path (OCTint) for imaging objects in distances of 5 mm to 10 mm from the chip edge, and another one with an external reference path (OCText) for use with conventional scan heads. Both OCT systems use integrated photodiodes and an external swept-frequency source. In our proof-of-concept experiments, we achieve a sensitivity of -64 dB (-53 dB for OCTint) and a dynamic range of 60 dB (53 dB for OCTint). The viability of the concept is demonstrated by imaging of biological and technical objects.