Binary Decision Diagrams (BDDs) are a data structure used to represent and manipulate Boolean functions symbolically. They are particularly effective as the algorithmic basis for symbolic model checkers, which use BDDs to encode sets and relations as Boolean functions and perform symbolic operations on them. The paper introduces ordered Binary Decision Diagrams (OBDDs), which impose an ordering constraint on the variables, ensuring a unique and reduced representation of any function. This ordering allows for efficient algorithms to reduce OBDDs to their unique form and implement a variety of operations on Boolean functions. The paper discusses the API for Boolean function manipulation, including operations for creating, manipulating, and analyzing Boolean functions. It also covers the implementation of these operations, focusing on depth-first traversals, memoization, and the use of unique tables to enforce reduction rules. The paper highlights the benefits of shared representations, where multiple functions are represented by a single, multi-rooted graph, and the use of complement edges to optimize operations like NOT. The paper explores variable ordering and reordering techniques, including dynamic variable ordering through sifting, which can significantly improve memory performance. It also surveys variants of BDDs, such as zero-suppressed BDDs (ZDDs) and partitioned BDDs, which are effective for specific applications like sparse sets and large state spaces. Additionally, the paper discusses representing non-Boolean functions, including functions over discrete and unbounded domains, using multi-terminal BDDs (MTBDDs) and Difference Decision Diagrams (DDDs). Overall, the paper provides a comprehensive overview of BDDs, their applications, and the techniques used to optimize their implementation and performance.Binary Decision Diagrams (BDDs) are a data structure used to represent and manipulate Boolean functions symbolically. They are particularly effective as the algorithmic basis for symbolic model checkers, which use BDDs to encode sets and relations as Boolean functions and perform symbolic operations on them. The paper introduces ordered Binary Decision Diagrams (OBDDs), which impose an ordering constraint on the variables, ensuring a unique and reduced representation of any function. This ordering allows for efficient algorithms to reduce OBDDs to their unique form and implement a variety of operations on Boolean functions. The paper discusses the API for Boolean function manipulation, including operations for creating, manipulating, and analyzing Boolean functions. It also covers the implementation of these operations, focusing on depth-first traversals, memoization, and the use of unique tables to enforce reduction rules. The paper highlights the benefits of shared representations, where multiple functions are represented by a single, multi-rooted graph, and the use of complement edges to optimize operations like NOT. The paper explores variable ordering and reordering techniques, including dynamic variable ordering through sifting, which can significantly improve memory performance. It also surveys variants of BDDs, such as zero-suppressed BDDs (ZDDs) and partitioned BDDs, which are effective for specific applications like sparse sets and large state spaces. Additionally, the paper discusses representing non-Boolean functions, including functions over discrete and unbounded domains, using multi-terminal BDDs (MTBDDs) and Difference Decision Diagrams (DDDs). Overall, the paper provides a comprehensive overview of BDDs, their applications, and the techniques used to optimize their implementation and performance.