In　this　paper,　we　design　a　multilevel　local　defect-correction　method　to　solve　the　non-selfadjoint　Steklov　eigenvalue　problems.　Since　the　computation　work　needed　for　solving　the　non-selfadjoint　Steklov　eigenvalue　problems　increases　exponentially　as　the　scale　of　the　problems　increase,　the　main　idea　of　our　algorithm　is　to　avoid　solving　large-scale　equations　especially　large-scale　Steklov　eigenvalue　problems　directly.　Firstly,　we　transform　the　non-selfadjoint　Steklov　eigenvalue　problem　into　some　symmetric　boundary　value　problems　defined　in　a　multilevel　finite　element　space　sequence,　and　some　small-scale　non-selfadjoint　Steklov　eigenvalue　problems　defined　in　a　low-dimensional　auxiliary　subspace.　Next,　the　local　defect-correction　method　is　used　to　solve　the　symmetric　boundary　value　problems,　then　the　difficulty　of　solving　these　symmetric　boundary　value　problems　is　further　reduced　by　decomposing　these　large-scale　problems　into　a　series　of　small-scale　subproblems.　Overall,　our　algorithm　can　obtain　the　optimal　error　estimates　with　linear　computational　complexity,　and　the　conclusions　are　proved　by　strict　theoretical　analysis　which　are　different　from　the　developed　conclusions　for　equations　with　the　Dirichlet　boundary　conditions.