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INTRODUCTION PART 1. INFINITE SMOOTHNESS OF SOLUTIONS CHAPTER 1. FUNDAMENTAL SOLUTIONS AND NONBOUNDED SOLUTIONS OF LINEAR HOMOGENEOUS DEGENERATE ELLIPTIC EQUATIONS 1.1. Basic definitions, hypergeometric functions 1.2. The case of finite degeneracy 1.3. The case of infinite degeneracy CHAPTER 2. SMOOTHNESS OF SOLUTIONS OF SEMILINEAR DEGENERATE ELLIPTIC EQUATIONS 2.1. Auxiliary results 2.2. Semilinear equations of HSrmander's type 2.3. Higher order semilinear hypoelliptic equations 2.4. Semilinear hypoelliptic equations of the Gilioli - Treves type 2.5. Equations with infinite degeneracy PART 2. ANALYTICITY OF SOLUTIONS CHAPTER 3. A NEW APPROACH FOR PROVING GEVREY REGULARITY OF SOLUTIONS OF ELLIPTIC EQUATIONS 3.1. Friedman's Lemmas 3.2. Weighted Holder spaces 3.3. Douglis-Nirenberg estimates 3.4. New argument in the proof of analyticity and Gevrey regularity of solutions CHAPTER 4. ANALYTICITY AND GEVREY REGULARITY OF SOLUTIONS OF SEMILINEAR EQUATIONS OF MIZOHATA TYPE 4.1. Uniform fundamental solutions and representation formula 4.2. Infinite smoothness of solutions 4.3. Analyticity and Gevrey regularity of solutions CHAPTER 5. ANALYTICITY AND GEVREY REGULARITY OF SOLUTIONS OF SEMILINEAR EQUATIONS OF THE GILIOLI - TREVES TYPE 5.1. A Model of Grushin's type equations 5.2. Equation of tile Gilioli - Treves type CHAPTER 6. ANALYTICITY AND GEVREY REGULARITY OF SOLUTIONS OF SEMILINEAR KOHN-LAPLACIAN ON THE HEISENBERG GROUP 6.1. Semilinear Kohn- Laplacian. 6.2. Fundamental solution of the Kohn - Laplacian 6.3. Oevrey regularityof solutions PART 3. GLOBAL PROPERTIES OF SOLUTIONS CHAPTER 7. EXISTENCE OF NONTRIVIAL SOLUTIONS TO BOUNDARY VALUE PROBLEMS FOR. SEMILINEAR DEGENERATE ELLIPTIC EQUATIONS 7.1. Embedding theorems of Sobolev's type 7.2. Theorems on existence of solutions CHAPTER 8. NON EXISTENCE OF NONTRIVIAL SOLUTIONS TO BOUNDARY VALUE PROBLEMS AND REGULARITY UP TO THE BOUNDARY 8.1 Generalized Pohozaev's identity and theorems on nonexistence of solutions 8.2. Regularity of solutions to boundary value problems up to the boundary BIBLIOGRAPHY LIST OF NOTATIONS AND ABBREVIATIONS ±à¼­ÊÖ¼Ç