  The hyperfine structure constants of the lowest $s$ and $p_{1/2}$ states of
superheavy elements Z=119 and Z= 120$^+$ are calculated using {\em ab initio}
approach. Core polarization and dominating correlation effects are included to
all orders. Breit and quantum electrodynamic effects are also considered.
Similar calculations for Cs, Fr, Ba$^+$ and Ra$^+$ are used to control the
accuracy. The dependence of the hyperfine structure constants on nuclear radius
is discussed.
