The question we consider in this talk is, which lens
spaces are obtainable by surgery on knots in the 3-sphere?
We first discuss this question for lens spaces of
type $(p,q)$ with small $p$ or $q$. Next, we will
particularly show that lens spaces containing Klein
bottles are unobtainable by Berge's surgery on doubly
primitive knots. We also give an algorithm to determine
whether a given lens space is obtainable by Berge's
surgery on doubly primitive knots.