We define a very general group manipulation idea and the corresponding stability concept of “absence-proofness”. In the first chapter, we analyze this concept in surplus sharing transferable utility games, exchange economies with private endowments and fair division problems. Solutions that are stable in our sense are core selections. We also show that it is weaker than population monotonicity in cooperative games and fair division problems, and very demanding for the allocation problems with private endowments. Particularly, the Walrasian allocation rule is not immune to manipulation. Also, it is the first external stability concept defined for fair division problems. In the second chapter, we work on cooperative stability in cost sharing of a minimum cost spanning tree and give a family of stable solutions that are responsive to the asymmetries in the cost data. Interpreting population monotonicity as a strong stability property, in the third chapter, we study population monotonicity in the fair division of indivisible goods where monetary compensations are allowed. We show that if there are more than three goods no efficient solution satisfies this property. For the two goods case we define hybrid solutions that are efficient and population monotonic.