Many classical problems in additive additive group theory andadditive number theory are direct problems, in which one starts with twosubsets A and B of a group, and tries to describe the structure of thesumset A+B consisting of all sums of elements of A and B. Bycontrast, in an inverse problem, one starts with a sumset A+B, andattempts to describe the structure of the underlying set a and B.This thesis studies some basic combinatorial problems ofan abelian group,especially those related to direct problemsand inverse problems.Contents of the research is divided intosix chapters.In chapter 1,we introduce some basic notations andconcepts, and summarize the backgroud and advancement ofthe research on direct problem and inverse problem of additive grouptheory and additive number theory.In chapter 2, we obtain a new alternative formulation ofKneser's theorem,which is obtained by applying the method ofset theory, i.e. |A+B|≥|A|+|B|-(?){|(A-a)∩(B-b)∩H|, |(A-a)∩(b-B)∩H|,|(A-a)∩[h-(B-b)∩H]∩H|}.where H=H(A+B)={g∈G|A+B+g=A+B}.This result have manyapplications. In chapter 3, we mainly include two generalizations ofKemperman's Structure Theorem (KST) for critical pairs(i.e.those finite subsets A and B of an abelian group with|A+B|≤|A|+|B|) through quasi-periodic decompositions and anew alternative formulation of Kneser's Theorem.In Chapter 4, we completely characterize some aspects ofquasi-periodic decomposition, which are the generalizations ofthe aspects obtained by D.Grynkiewicz. For example,we provethat if A_1∪A_0 and A′_1∪A′_0 are both quasi-periodic decompositions of asubset A of an abelian group G,with A_1 H-periodic and A′_1L-periodic,then one of the following conclusions holds:(ⅰ) A_0=A′_0, A_1=A′_1;(ⅱ) there exists a subset K of some H∩L-coset such that A∪Kis (H+L)-periodic;(ⅲ) L is a proper subgroup of H and A_0 is L- quasi-periodic;(ⅳ) H is a proper subgroup of L and A′_0 is H- quasi-periodic.In addition,we give a new united explanation of Kemperman'scurious result and extend Kemperman's result.In chapter 5, we refine the aspects of the sumsetsum from n=≤h(S) (S),which is established by Gao Wei Dong,where Sis asequence of an abelian group,we give a new simple proof ofthe results of Chuang Peng,which is related to Olson's constant and we obtain some new upper bounds on the Frobeniusnumber with some methods of additive group theory.In chapter 6, we improve G.Zemor's result about theadditive base; and determine s_p(G) and t_p(G) in special cases.