We　present　a　spectral　approach　to　design　approximation　algorithms　for　network　design　problems.　We　observe　that　the　underlying　mathematical　questions　are　the　spectral　rounding　problems,　which　were　studied　in　spectral　sparsification　and　in　discrepancy　theory.　We　extend　these　results　to　incorporate　additional　nonnegative　linear　constraints,　and　show　that　they　can　be　used　to　significantly　extend　the　scope　of　network　design　problems　that　can　be　solved.　Our　algorithm　for　spectral　rounding　is　an　iterative　randomized　rounding　algorithm　based　on　the　regret　minimization　framework.　In　some　settings,　this　provides　an　alternative　spectral　algorithm　to　achieve　constant　factor　approximation　for　the　classical　survivable　network　design　problem,　and　partially　answers　a　question　of　Bansal　about　survivable　network　design　with　concentration　property.　We　also　show　many　other　applications　of　the　spectral　rounding　results,　including　weighted　experimental　design　and　spectral　network　design.　©　2022　Society　for　Industrial　and　Applied　Mathematics.