The double-beta decay of Cd116 has been investigated with the help of radiopure enriched Cd116WO4 crystal scintillators (mass of 1.162 kg) at the Gran Sasso underground laboratory. The half-life of Cd116 relative to the 2ν2β decay to the ground state of Sn116 was measured with the highest up-to-date accuracy as T1/2=(2.63-0.12+0.11)×1019 yr. A new improved limit on the 0ν2β decay of Cd116 to the ground state of Sn116 was set as T1/2≥2.2×1023 yr at 90% C.L., which is the most stringent known restriction for this isotope. It corresponds to the effective Majorana neutrino mass limit in the range ⟨mν⟩≤(1.0–1.7) eV, depending on the nuclear matrix elements used in the estimations. New improved half-life limits for the 0ν2β decay with majoron(s) emission, Lorentz-violating 2ν2β decay, and 2β transitions to excited states of Sn116 were set at the level of T1/2≥1020–1022 yr. New limits for the hypothetical lepton-number violating parameters (right-handed currents admixtures in weak interaction, the effective majoron-neutrino coupling constants, R-parity violating parameter, Lorentz-violating parameter, heavy neutrino mass) were set.