We construct a semiorthogonal decomposition of the derived category of coherent sheaves on a quadric fibration consisting of several copies of the derived category of the base of the fibration and the derived category of coherent sheaves of modules over the sheaf of even parts of the Clifford algebras on the base corresponding to this quadric fibration. This implies that the universal sheaf of even parts of CLifford algebras is the Homologically Projectively Dual variety to the double Veronese space and gives a description of the derived category of coherent sheaves on a complete intersection of any number of quadrics.