Powder bed fusion (PBF) is a popular additive manufacturing (AM) process with wide applications in key industrial sectors, including aerospace, automotive, healthcare, defence. However, a deficiency of PBF is its low quality of surface finish. A number of PBF process variables and other factors (e.g. powders, recoater) can influence the surface quality. It is of significant importance to measure and characterise PBF surfaces for the benefits of process optimisation, product performance evaluation and also product design. A state-of-the-art review is given to summarise the current research work on the characterisation of AM surfaces, particularly PBF surfaces. It is recognised that AM processes are different from conventional manufacturing processes and their produced surface topographies are different as well. In this paper, the surface characterisation framework is updated to reflect the unique characteristics of PBF processes. The surface spatial wavelength components and other process signature features are described and their production mechanisms are elaborated. A bespoke surface characterisation procedure is developed based on the updated framework. The robust Gaussian regression filter and the morphological filters are proposed to be used for the separation of the waviness component due to their robustness. The watershed segmentation is enhanced to extract globules from the residual surface. Two AM components produced by electron beam melting (EBM) and selective laser melting (SLM), are measured and characterised by the proposed methodology. Both of the two filters are qualified for the extraction of melted tracks. The watershed segmentation can enable the extraction of globules. The standard surface texture parameters of different surface wavelength components are compared. A set of bespoke parameters are intentionally developed to offer a quantitative evaluation of the globules.