The emergence of high-speed networks such as those with ATM integrates large numbers of services with a wide range of characteristics. Admission control is a prime instrument for controlling congestion in the network. As part of connection services to an ATM system, the Connection Admission Control (CAC) algorithm decides if another call or connection can be admitted to the Broadband Network. The main task of the CAC is to ensure that the broadband resources will not saturate or overflow within a very small probability. It limits the connections and guarantees Quality of Service for the new connection. The algorithm for connection admission is crucial in determining bandwidth utilisation efficiency. With statistical multiplexing more calls can be allocated on a network link, while still maintaining the Quality of Service specified by the connection with traffic parameters and type of service. A number of algorithms for admission control for Broadband Services with ATM Networks are described and compared for performance under different traffic loads. There is a general description of the ATM Network as an introduction. Issues to do with source distributions and traffic models are explored in Chapter 2. Chapter 3 provides an extensive presentation of the CAC algorithms for ATM Broadband Networks. The ideas about the Effective Bandwidth are reviewed in Chapter 4, and a different approach to admission control using online measurement is presented in Chapter 5. Chapter 6 has the numerical evaluation of four of the key algorithms, with simulations. Finally Chapter 7 has conclusions of the findings and explores some possibilities for further work.