This chapter presents recent results on context-free languages, focusing on topics not covered in existing textbooks. Context-free languages and grammars were initially designed to formalize the grammatical properties of natural languages and are now widely used for describing the syntax of programming languages. The chapter covers recent developments in algebraic treatments of these topics, including investigations into open problems such as the equivalence of deterministic pushdown automata and the existence of principal cones with non-generator languages. The chapter is divided into several sections: 1. **Introduction**: Discusses the historical context and recent research trends in context-free languages. 2. **Iteration**: Introduces iteration lemmas, the interchange lemma, and degeneracy, providing examples and proofs. 3. **Looking for nongenerators**: Explores the concept of generators and nongenerators in context-free languages, including their relationship with determinism and the intersection of principal cones. 4. **Context-free groups**: Provides an overview of context-free groups, their presentation, word problems, and global characterizations, along with Cayley graphs and ends. Key topics include: - **Algebraic Development**: The existence of an invariant for context-free languages, the Hotz group. - **Iteration Lemmas**: Proofs and applications, including the interchange lemma and its use in proving languages are not context-free. - **Nongenerators**: Properties and implications for the structure of context-free languages. - **Context-free Groups**: Theoretical aspects and connections to other areas of mathematics. The chapter also includes examples and proofs to illustrate the concepts and results discussed.This chapter presents recent results on context-free languages, focusing on topics not covered in existing textbooks. Context-free languages and grammars were initially designed to formalize the grammatical properties of natural languages and are now widely used for describing the syntax of programming languages. The chapter covers recent developments in algebraic treatments of these topics, including investigations into open problems such as the equivalence of deterministic pushdown automata and the existence of principal cones with non-generator languages. The chapter is divided into several sections: 1. **Introduction**: Discusses the historical context and recent research trends in context-free languages. 2. **Iteration**: Introduces iteration lemmas, the interchange lemma, and degeneracy, providing examples and proofs. 3. **Looking for nongenerators**: Explores the concept of generators and nongenerators in context-free languages, including their relationship with determinism and the intersection of principal cones. 4. **Context-free groups**: Provides an overview of context-free groups, their presentation, word problems, and global characterizations, along with Cayley graphs and ends. Key topics include: - **Algebraic Development**: The existence of an invariant for context-free languages, the Hotz group. - **Iteration Lemmas**: Proofs and applications, including the interchange lemma and its use in proving languages are not context-free. - **Nongenerators**: Properties and implications for the structure of context-free languages. - **Context-free Groups**: Theoretical aspects and connections to other areas of mathematics. The chapter also includes examples and proofs to illustrate the concepts and results discussed.