User-Defined Functions (UDFs) are custom functions created by users or developers in programming languages or database systems to perform specific tasks or calculations. These functions extend the functionality of a programming language or a database management system (DBMS) and can be categorized into several types:

1. Scalar UDFs:    Scalar UDFs return a single value for each input parameter. They are commonly used for calculations and transformations on individual data values. Examples include mathematical calculations, string manipulations, and date/time conversions.

2. Table-Valued UDFs (TVFs):    TVFs return a table as their output. They can be used to encapsulate complex queries or operations that produce a set of rows. TVFs are helpful for breaking down complex queries into manageable parts and for reusing query logic. They can be used in SELECT statements just like database tables.

3. Inline Table-Valued UDFs:    These are similar to TVFs but are optimized differently by database engines. Inline TVFs return an entire table as part of the query execution plan and are generally more efficient for simple queries.

4. Multi-Statement Table-Valued UDFs:    These TVFs are used when the query logic is more complex and involves multiple statements. They are useful for encapsulating complex data processing operations.

5. Aggregate UDFs:   Aggregate UDFs are used for performing custom aggregation operations on a set of values, such as calculating the median or mode of a dataset. They extend the built-in aggregation functions provided by DBMSs.

6. CLR (Common Language Runtime) UDFs:    CLR UDFs are used in database systems like Microsoft SQL Server. They allow developers to write UDFs using .NET languages like C# or Visual Basic, providing a powerful way to create custom functions with complex logic.

7. Custom UDFs in Programming Languages:    In programming languages like Python, JavaScript, or Java, developers can create custom functions tailored to their specific application requirements. These functions can perform various tasks, such as data processing, calculations, and logic execution.

UDFs offer the flexibility to extend the functionality of programming languages and database systems to meet specific application needs. They improve code reusability, readability, and maintainability by encapsulating logic into reusable units, making development more efficient and scalable. However, it's essential to use UDFs judiciously, as poorly designed functions can impact performance and maintainability.