  Upper and lower bounds on the largest number of weights in a cyclic code of
given length, dimension and alphabet are given. An application to irreducible
cyclic codes is considered. Sharper upper bounds are given for the special
cyclic codes (called here strongly cyclic), {whose nonzero codewords have
period equal to the length of the code}. Asymptotics are derived on the
function $\Gamma(k,q),$ {that is defined as} the largest number of nonzero
weights a cyclic code of dimension $k$ over $\F_q$ can have, and an algorithm
to compute it is sketched. The nonzero weights in some infinite families of
Reed-Muller codes, either binary or $q$-ary, as well as in the $q$-ary Hamming
code are determined, two difficult results of independent interest.
