You can check if PostgreSQL is running and accepting connections using various built-in utilities like `pg_isready`, the interactive terminal `psql`, or by checking the status of the PostgreSQL service itself. These methods allow you to determine whether the database server is operational, ready to handle incoming connections, and accept connections from outside localhost. Additionally, programming languages like Python and Java can be used to test PostgreSQL connections programmatically, providing a convenient way to automate connectivity checks or integrate them into larger applications for real-time monitoring and troubleshooting.