(1) The main features of the ten-year cycle are the regularity of the period and the irregularity of the amplitude of the oscillations; these features are obvious in data on the lynx cycle, and in the correlogram and periodogram calculated from the data. (2) A statistical model is proposed for the analysis of the 10-year cycle which takes these features into account by including both a strictly periodic term and an autoregressive term depending on the size of the population in the previous year; it is shown that this model fits the data better than the second-order autoregressive model used by Moran (1953). (3) All data on the ten-year cycle in Canada have the same period of about 9.63 years. The other parameters in the model can be estimated by the method of least squares, and estimates of the phase and the average amplitude of the oscillations can then be obtained. (4) This method is used to analyse data on the fur-bearing mammals of Canada between 1751 and 1969. A 10-year cycle exists during at least part of this time in the following species: coyote, fisher, red fox, lynx, marten, mink, muskrat, skunk, wolf, wolverine and snowshoe rabbit. A cycle has also been confirmed in the Atlantic salmon. There is a tendency for the cycle to be most pronounced in the midwest of Canada and to become weaker and later as one moves away from this region. A similar cycle exists in the taiga zone of Russia. (5) The simplest theory is that the cycle in all other species is caused, directly or indirectly, by the cycle in the snowshoe rabbit. The food habits of all the cyclic species are reviewed with this theory in mind. There is considerable difficulty in linking some of the cyclic species convincingly with the snowshoe rabbit, but this is nevertheless still thought to be the most likely explanation since no cyclic meteorological factor has been discovered.