Mobile broadband (MBB) penetration has deepened globally over the last twenty years. This is largely due to the adoption of smart devices, improved mobile communications network coverage, and the perpetual drive to develop ever faster mobile and wireless communication technologies. However, information on the quality of service (QoS) delivered by MBB operators to the end users remains an issue of concern. This has driven independent researchers and mobile communication industry regulators to develop methodologies for independent and unbiased evaluation of the QoS offered by MBB networks. This paper provides a detailed review of MBB adoption and penetration across several regions of the world. It also includes the existing methodologies for evaluating the performance of MBB systems as experienced by the end user. Specifically, methodologies such as the drive and walk tests, crowd-sourced mobile device-based methods and the software applications they employ, and the dedicated measurement testbeds are reviewed. Based on this, the challenges of adopting each of the methods are discussed in order to make a case for the development of more robust, partially autonomous and scalable MBB measurement platforms for the future.