Many and varied claims have been made over many years for a relationship between weather or climate and solar variations, notably sunspot cycles. Those relating primarily to the single and double sunspot cycles (of about 11- and 22-year quasi-periodicities) are critically reviewed in the light of what is known about solar variations, the observed variability of weather and climate, and possible physical connections between the two. Various pitfalls in the application or lack of application of statistics to the problem are discussed and illustrated from the literature. Following a survey of the literature it is concluded that despite the great number of recent papers on the subject, little convincing evidence has yet been produced for real correlations between sunspot cycles and the weather/climate on the 11- and 22-year time scales, although evidence for correlations with solar events on time scales of days appears to exist. The state of the literature in this particularly controversial area must raise doubts as to the prevailing standards of objectivity and critical analysis in other areas of science as well. Clearly, in the case of sun-weather relationships, further research requires much higher standards of objectivity, with the rigorous and critical application of statistics, and step by step investigation of hypothetical mechanisms. This criticism is not addressed to the recent studies of apparently significant correlations between certain meteorological indices and the passage of interplanetary magnetic sector boundaries; however, the relevance of such correlations to time scales of climatic interest has yet to be demonstrated.