Consider　a　varying-coefficient　single-index　model　which　consists　of　two　parts:　the　linear　part　with　varying　coefficients　and　the　nonlinear　part　with　a　single-index　structure,　and　are　hence　termed　as　varying-coefficient　single-index　models.　This　model　includes　many　important　regression　models　such　as　single-index　models,　partially　linear　single-index　models,　varying-coefficient　model　and　varying-coefficient　partially　linear　models　as　special　examples.　In　this　paper,　we　mainly　study　estimating　problems　of　the　varying-coefficient　vector,　the　nonparametric　link　function　and　the　unknown　parametric　vector　describing　the　single-index　in　the　model.　A　stepwise　approach　is　developed　to　obtain　asymptotic　normality　estimators　of　the　varying-coefficient　vector　and　the　parametric　vector,　and　estimators　of　the　nonparametric　link　function　with　a　convergence　rate.　The　consistent　estimator　of　the　structural　error　variance　is　also　obtained.　In　addition,　asymptotic　pointwise　confidence　intervals　and　confidence　regions　are　constructed　for　the　varying　coefficients　and　the　parametric　vector.　The　bandwidth　selection　problem　is　also　considered.　A　simulation　study　is　conducted　to　evaluate　the　proposed　methods,　and　real　data　analysis　is　also　used　to　illustrate　our　methods.　(C)　2010　Elsevier　Inc.　All　rights　reserved.