The main law of the evolution of the Earth’s climate is the cyclical nature of global changes in the latter. One of the possible explanations for the cyclical nature of global climate changes is provided by the astrophysical model of fluctuations in the insolation of the Earth’s surface by solar radiation. Modern climate change is mainly associated with variations in the magnetic activity of the Sun, one of the main proxies of which are sunspots. The decrease in the number of sunspots coincides with the epochs of cooling on the Earth, while during the maximum number of sunspots warming is observed. The paper reviews cosmogenic reconstructions of long-term variations in the Sun’s magnetic activity (large minima and large maxima) during the Holocene (last 12,000 years). The accidental appearance of large minima and maxima can to some extent be reproduced by modern models of a turbulent dynamo with a stochastic drive. An important key to studying the impact of solar activity variations on the Earth’s climate is the Maunder minimum (late 17th century), during which extremely little sunspots were observed. Applying the method of analysis of rare events to these observations led researchers to conclude that the appearance of sunspots at the Maunder minimum was characterized by a weak amplitude of 22 years. The concept of continuity of magnetic cycles at this time is also confirmed by measurements of cosmogenic radionuclides in natural terrestrial archives. Therefore, today it is believed that during the Maunder minimum, the cyclic magnetic activity of the Sun did not stop, although the amplitude of the cycles was quite low. In the αΩ-dynamo model, this may be due to the fact that the magnitude of the magnetic induction of the toroidal field excited by radial differential rotation in the solar convection zone at this time did not reach the threshold value required for lifting magnetic power tubes on the solar surface (nonlinear dynamo mode). Possible physical mechanisms describing the suppression of the dynamo process at intervals when no sunspots were observed are analyzed. A scenario for explaining the north-south asymmetry of magnetic activity during the Maunder minimum is proposed. A key role in the proposed scenario is played by the special nature of the internal rotation of the Sun, revealed as a result of helioseismological experiments. The modern grand maximum of solar activity, which began in the 1940s, has ceased after solar cycle 23, and activity of the Sun seems to be returning to its normal moderate level.