Summary: It has been known since 1996 [J.-Y. Cai et al., Math. Syst. Theory 29, No. 3, 245–258 (1996; Zbl 0858.94033)] that the sequence of Hamming weights \(\{\mathrm{wt}(f_n(x_1, \ldots, x_n)) \}\), where \(f_n\) is a symmetric Boolean function of degree \(d\) in \(n\) variables, satisfies a linear recurrence with integer coefficients. F. N. Castro and L. A. Medina [Electron. J. Comb. 18, No. 2, Research Paper P8, 21 p. (2011; Zbl 1250.11102)] used this result to show that a conjecture of T. W. Cusick et al. [IEEE Trans. Inf. Theory 54, No. 3, 1304–1307 (2008; Zbl 1306.94041)] about when an elementary symmetric Boolean function can be balanced is true for all sufficiently large \(n\). Quite a few papers have been written about this conjecture, but it is still not completely settled. Recently, Y. Guo et al. [“Recent results on balanced symmetric Boolean functions” (2015; doi:10.1109/TIT.2015.2455052)] proved that the conjecture is equivalent to the statement that all balanced elementary symmetric Boolean functions are trivially balanced. This motivates the further study of the weights of symmetric Boolean functions \(f_n\). In this paper we prove various new results on the trivially balanced functions. We also determine a period (sometimes the minimal one) for the sequence of weights \(\mathrm{wt}(f_n)\) modulo any odd prime \(p\), where \(f_n\) is any symmetric function, and we prove some related results about the balanced symmetric functions.