It is generally believed that non-Hermiticity can transform Weyl semimetals into Weyl-exceptional-ring semimetals. However, this belief is from the systems without skin effect. We investigate the non-Hermitian Weyl semimental and its Floquet engineering in a system with skin effect, which breaks the bulk-boundary correspondence in its Hermitian counterpart. It is found in both the static and periodically driven cases that the skin effect makes this general belief no longer valid. We discover that exotic non-Hermitian topological matters, e.g., a composite phase of Weyl semimetal and topological insulator with the coexisting Fermi arc and chiral boundary states, a widely tunable Hall conductivity with multiple quantized plateaus, and a Weyl semimetal with anomalous Fermi arcs formed by the crossing of gapped bound state, can be generated by the Floquet engineering. Revealing the leading role of the skin effect in determining the feature of a semimental, our result supplies a useful way to artificially synthesize exotic non-Hermitian Weyl semimetals by periodic driving.

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• Topological phase transition
• Floquet systems
• Weyl semimetal