  Double beta decay of $^{116}$Cd has been investigated with the help of
radiopure enriched $^{116}$CdWO$_4$ crystal scintillators in the experiment
Aurora. The half-life of $^{116}$Cd relatively to the 2$\nu$2$\beta$ decay of
$^{116}$Cd to the ground level of $^{116}$Sn is measured with the highest
up-to-date accuracy as $T_{1/2}$ = [2.69 $\pm$ 0.02 (stat.) $\pm$ 0.14 (syst.)]
$\times$ 10$^{19}$ yr. A new improved limit on the 0$\nu$2$\beta$ decay of
$^{116}$Cd to the ground state of $^{116}$Sn is set as $T_{1/2}\geq 2.4 \times
10^{23}$ yr at 90\% C.L., that corresponds to the effective Majorana neutrino
mass limit in the range $\langle$$m_\nu$$\rangle$ $\le$ $(1.1-1.6)$ eV,
depending on the nuclear matrix elements used in the estimations. New improved
limits on other $2\beta$ processes in $^{116}$Cd (decays with majoron emission,
transitions to excited levels of $^{116}$Sn) were set at the level of
$T_{1/2}\geq 10^{21}-10^{22}$ yr.
