A　new　spectral　method　based　on　the　hybrid　formula　system　of　the　state-vector　and　Legendre　polynomials　are　used　to　derive　the　explicit　expressions　of　circumferential　guided　waves　in　the　anisotropic　multilayer　composite　cylinders　with　an　arbitrary　lay-up.　In　this　hybrid　method,　the　state　vector　was　utilized　to　transform　the　wave　equation,　boundary　conditions,　and　interface　continuous　conditions　in　the　cylinder　coordinate　system　to　form　the　dispersion　equation　in　terms　of　the　state　matrix.　Then,　following　the　principle　of　the　Galerkin　method,　projecting　the　dispersion　state　matrix　equations　onto　the　weight　functions　consisting　of　Legendre　polynomials,　the　system　of　algebraic　equation　is　obtained.　The　dispersion　curves　and　mode　shape　can　be　obtained　by　solving　the　eigenvalues　and　eigenvectors　of　the　system　of　algebraic　equations.　This　hybrid　method　uses　the　orthogonal　and　recursive　properties　of　the　Legendre　polynomial　to　simplify　the　integral　expressions　involved　in　all　the　matrices　and　yields　the　closed-form　solutions.　Numerical　verification,　including　the　cases　of　isotropic　and　anisotropic　viscoelastic/elastic　multilayer　cylinders,　was　conducted　to　evaluate　the　performance　of　the　proposed　method.　The　results　show　that　the　proposed　method　overcomes　the　cumbersome　of　the　traditional　Legendre　polynomial　method　to　treat　the　interface　displacement　and　stress　continuity.　The　results　also　confirm　that　the　hybrid　method　has　the　exponential　convergence.　©　2020　Elsevier　Ltd