Quantum entanglement is one essential element to characterize many-body quantum systems. However, so far, the entanglement measures mainly restrict to Hermitian systems. Here, we propose a natural extension of entanglement and Renyi entropies to non-Hermitian quantum systems. We demonstrate the generic entanglement and Renyi entropies capture the correct entanglement properties in non-Hermitian critical systems, where the low-energy properties are governed by the non-unitary conformal field theories (CFTs). We find excellent agreement between the numerical extrapolation of the negative central charges from the generic entanglement/Renyi entropy and the non-unitary CFT prediction. Furthermore, we apply the generic entanglement/Renyi entropy to symmetry-protected topological phases with non-Hermitian perturbations. We find the generic $n$-th Renyi entropy captures the expected entanglement property, whereas the traditional Renyi entropy can exhibit unnatural singularities due to its improper definition.
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