Thegoalofthisbookistoinvestigatethebehaviourofweaksolutionstotheelliptic transmisssion problem in a neighborhood of boundary singularities: angular and conic points or edges. We consider this problem both for linear and quasi-linear (very little studied) equations. In style and methods of research,this book is close to our monograph [14] together with Prof. V. Kondratiev. The book consists of an Introduction, seven chapters, a Bibliography and Indexes. Chapter 1 is of auxiliary character. We recall the basic de?nitions and properties of Sobolev spaces and weighted Sobolev-Kondratiev spaces. Here we recall also the well-known Stampacchia's Lemma and derive a generalization for the solution of the Cauchy problem - the Gronwall-Chaplygin type inequality. Chapter 2 deals with the eigenvalue problem for m-Laplace-Beltrami op- ator. By the variational principle we prove a new integro-di?erential Friedrichs- Wirtinger type inequality. This inequality is the basis for obtaining of precise exponents of the decreasing rate of the solution near boundary singularities. Chapter 3 deals with the investigation of the transmission problem for linear elliptic second order equations in the domains with boundary conic point. Chapter 4 is devoted to the transmission problem in conic domains with N di?erent media for an equation with the Laplace operator in the principal part. Chapters 5, 6 and 7 deal with the investigation of the transmission problem forquasi-linearellipticsecondorderequationsinthe domainswithboundaryconic point (Chapters 5-6) or with an edge at the boundary of a domain.