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Aims: The 11-year solar cycle (Schwabe cycle) is the dominant pattern of solar magnetic activity reflecting the oscillatory dynamo mechanism in the Sun’s convection zone. Solar cycles have been directly observed since 1700, while indirect proxy data suggest their existence over a much longer period of time, but generally without resolving individual cycles and their continuity. Here we reconstruct individual solar cycles for the last millennium using recently obtained ¹⁴C data and state-of-the-art models. Methods: Starting with the ¹⁴C production rate determined from the so far most precise measurements of radiocarbon content in tree rings, solar activity was reconstructed in the following three physics-based steps: (1) correction of the ¹⁴C production rate for the changing geomagnetic field; (2) computation of the open solar magnetic flux; and (3) conversion into sunspot numbers outside of grand minima. All known uncertainties, including both measurement and model uncertainties, were straightforwardly accounted for by a Monte-Carlo method. Results: Cyclic solar activity is reconstructed for the period 971–1900 (85 individual cycles) along with its uncertainties. This more than doubles the number of solar cycles known from direct solar observations. We found that the lengths and strengths of well-defined cycles outside grand minima are consistent with those obtained from the direct sunspot observations after 1750. The validity of the Waldmeier rule (cycles with fast-rising phase tend to be stronger) is confirmed at a highly significant level. Solar activity is found to be in a deep grand minimum when the activity is mostly below the sunspot formation threshold for about 250 years. Therefore, although considerable cyclic variability in ¹⁴C is seen even during grand minima, individual solar cycles can hardly be reliably resolved therein. Three potential solar particle events, ca. 994, 1052, and 1279 AD, are shown to occur around the maximum phases of solar cycles. Conclusions: A new approximately 1000-year-long solar activity reconstruction, in the form of annual (pseudo) sunspot numbers with the full assessment of all known uncertainties, is presented based on new high-precision Δ¹⁴C measurements and state-of-the-art models, more than doubling the number of individually resolved solar cycles. This forms a solid basis for new, more detailed studies of solar variability.