The paper presents a survey of the mathematical theory of L systems, which are used to model the development of simple filamentous organisms. L systems are defined as linear arrays of finite automata, and later reformulated into grammar-like constructs. The theory of L systems is now a branch of formal language theory, and this paper discusses various aspects of L systems, including their language generating power, structural constraints, and growth functions. The paper begins by defining L schemes and L systems, which are used to generate languages. It then discusses different ways of extracting languages from L systems, including exhaustive and selective approaches. The paper also explores the relationship between L systems and other formal language theories, such as context-free and context-sensitive grammars. The paper then discusses the growth functions of L systems, which describe how the length of words generated by the system increases over time. It introduces the concept of DOL growth, which is a type of growth function associated with DOL systems. The paper also discusses the equivalence, analysis, and synthesis of DOL growth functions. The paper concludes by discussing various open problems in the theory of L systems, including the classification of growth types and the decidability of certain problems. It also highlights the importance of L systems in modeling biological processes and their potential applications in other areas of computer science.The paper presents a survey of the mathematical theory of L systems, which are used to model the development of simple filamentous organisms. L systems are defined as linear arrays of finite automata, and later reformulated into grammar-like constructs. The theory of L systems is now a branch of formal language theory, and this paper discusses various aspects of L systems, including their language generating power, structural constraints, and growth functions. The paper begins by defining L schemes and L systems, which are used to generate languages. It then discusses different ways of extracting languages from L systems, including exhaustive and selective approaches. The paper also explores the relationship between L systems and other formal language theories, such as context-free and context-sensitive grammars. The paper then discusses the growth functions of L systems, which describe how the length of words generated by the system increases over time. It introduces the concept of DOL growth, which is a type of growth function associated with DOL systems. The paper also discusses the equivalence, analysis, and synthesis of DOL growth functions. The paper concludes by discussing various open problems in the theory of L systems, including the classification of growth types and the decidability of certain problems. It also highlights the importance of L systems in modeling biological processes and their potential applications in other areas of computer science.