Photonic methods of radio-frequency waveform generation and processing can provide performance advantages and flexibility over electronic methods due to the ultrawide bandwidth offered by the optical carriers. However, bulk optics implementations suffer from the lack of integration and slow reconfiguration speed. Here we propose an architecture of integrated photonic radio-frequency generation and processing and implement it on a silicon chip fabricated in a semiconductor manufacturing foundry. Our device can generate programmable radio-frequency bursts or continuous waveforms with only the light source, electrical drives/controls and detectors being off-chip. It modulates an individual pulse in a radio-frequency burst within 4 ns, achieving a reconfiguration speed three orders of magnitude faster than thermal tuning. The on-chip optical delay elements offer an integrated approach to accurately manipulating individual radio-frequency waveform features without constraints set by the speed and timing jitter of electronics, and should find applications ranging from high-speed wireless to defence electronics.