To build a forum application with PHP, you can follow these steps:

1. Set up the database: Create a database to store forum data and tables to hold forum categories, topics, posts, and users.

2. Create a registration and login system: Implement a user registration and login system to allow users to create accounts and log in to the forum.

3. Design the forum layout: Create the layouts for displaying forum categories, topics, and posts. Use HTML, CSS, and PHP to construct the forum pages.

4. Implement CRUD functionality: Add functionality to create, retrieve, update, and delete forum categories, topics, and posts. Use PHP and SQL queries to interact with the database.

5. Add user roles and permissions: Implement different user roles such as administrators, moderators, and regular users. Define their permissions and restrict certain actions based on user roles.

6. Add features like notifications and messaging: Implement features such as email notifications for new posts, private messaging between users, and the ability to follow specific topics or users.

7. Implement search functionality: Allow users to search for specific topics or posts using keywords or filters. Use SQL queries to retrieve relevant data from the database.

8. Implement pagination: If there are a large number of topics or posts, implement pagination to display them in smaller chunks. Use PHP to calculate the number of pages and retrieve the appropriate data from the database.

9. Implement user profile pages: Create user profile pages where users can view and edit their profile information, such as their username, avatar, and bio.

10. Implement security measures: Use techniques like input validation and parameterized queries to prevent SQL injection and cross-site scripting (XSS) attacks. Also, hash and salt passwords to ensure user account security.

11. Implement moderation features: Add functionality for moderators and administrators to manage and moderate the forum, such as deleting inappropriate posts or banning users.

12. Test and debug: Test the forum application thoroughly to identify and fix any bugs or issues. Use tools like debugging software and error logs to track down and resolve any problems.

13. Deploy the forum application: Once the forum application is working correctly, deploy it to a web server so that users can access and use it.

By following these steps, you can create a functional forum application using PHP. Remember to regularly update and maintain the application to address any security vulnerabilities and provide a smooth user experience.