Library APIs (Application Programming Interfaces) are interfaces that allow developers to access the functionalities and resources of a software library. A software library is a collection of pre-built code modules that provide specific functions or services to facilitate the development of software applications.

Library APIs define the methods, classes, data types, and parameters that developers can use to access the library's functions. APIs act as intermediaries between the application logic written by developers and the core code of the library. They provide a standardized way to access the library's services without developers needing to understand the internal structure of the library.

Examples of library APIs could include:

1. Graphics library APIs: These allow developers to create graphics and animations in their applications. An example is the OpenGL API for 3D graphics.

2. Network library APIs: These offer functions for communication over networks, such as sending and receiving data over the internet. An example is the HTTP API used by web browsers and other applications to communicate with web servers.

3. Database library APIs: These facilitate access to databases for storing, retrieving, and manipulating data. Examples include the APIs of SQL databases like MySQL or PostgreSQL.

4. Mathematical library APIs: These provide mathematical functions and operations for complex calculations. Examples are the mathematical functions in Python or the BLAS API for numerical computations.

Developers can use library APIs to leverage functionalities developed by experienced developers or teams, rather than having to implement these features from scratch. This speeds up development, reduces code effort, and improves code quality by reusing proven solutions.