This dissertation consists of four theoretical essays.In all essays two-sided matching is a key factor.The first essay investigates the efficiency，strategy-proofness and monotonicity for agents with multi-unit demand in many-to-one matchings.The second essay deals with the blocking lemma and strategy-proofness in many-to-many matchings.　　The third essay studies the stability of many-to-many matching with max-min preferences.The last essay analyzes the blocking lemma and group incentive compatibility for matching with contracts.　　This first essay proposes the quota-saturability condition.It is shown that the maxmin preference criterion and the quota-saturability condition together guarantee weak Pareto efficiency and strategy-proofness for agents with multi-unit demand in many-to-one matchings.Moreover,I introduce a new notion of max-min criterion，called Wmax-min criterion，which together with the quota-saturability condition，ensures that the deferred acceptance algorithm is not only weakly Pareto efficient and strategy-proof,but also monotone for agents on the proposing side.　　This second essay considers the incentive compatibility in many-to-many two-sided matching problems.I first show that the Blocking Lemma holds for a many-to-many matching model under the max-min preference criterion and quota-saturability condition introduced in this paper.This result extends the Blocking Lemma for one-to-one matching and for many-to-one matching to many-to-many matching problem.It is then shown that the deferred acceptance mechanism is strategy-proof for agents on the proposing side under the max-min preference criterion and quota-saturability condition.Neither the Blocking Lemma nor the incentive compatibility can be guaranteed if the preference condition is weaker than the max-min criterion.　　This third essay investigates the two-sided many-to-many matching problem，where every agent has max-min preference.The equivalence between the pairwise-stability and the setwise-stability is obtained.It is shown that the pairwise-stability implies the strong corewise-stability and the former may be strictly stronger than the latter.I also show that the strong core may be a proper subset of the core.The deferred acceptance algorithm yields a pairwise-stable matching.Thus the set of stable matchings (in all four senses) is non-empty.　　This last essay considers a general class of two-sided many-to-one matching markets,so-called matching markets with contracts.I study the blocking lemma and group incentive compatibility for this class of matching markets.I first show that the blocking lemma for matching with contracts holds if hospitals preferences satisfy substitutes and the law of aggregate demand.The blocking lemma for one-to-one matching and that for many-to-one matching are special cases of this result.Then，as an immediate consequence of the blocking lemma，I show that the doctor-optimal stable mechanism is group strategy-proof for doctors if hospitals preferences satisfy substitutes and the law of aggregate demand.Hatfield and Kojima (2009) originally obtain this result by skillfully using the strategy-proofness of the doctor-optimal stable mechanism.In this paper I provide a different proof for the group incentive compatibility by applying the blocking lemma.