One of the key driving research questions in our work on Auslan and British Sign Language has been: sign languages are natural languages, but what *kind* of languages are they? In particular, how much does the grammatical organisation of specific sign languages resemble other signed and spoken languages, and how much does it differ (e.g., Cormier et al., 2010; Johnston et al., 2015)? What factors account for these similarities and differences (e.g., Johnston & Schembri, 2010)? In this presentation, I will revisit the so-called ‘paradox of sign language morphology’ (Aronoff et al., 2005). This refers to the fact that, despite being relatively ‘young’ languages, sign languages have characteristics of their grammatical organisation that seem unexpected. For example, many sign languages have indicating verbs (Fenlon et al., 2018). These verbs use spatial alternations to signal agent and patient roles and have been widely analysed as marking person agreement (Pfau et al., 2018). Agreement, however, is generally believed to take many centuries to emerge through grammaticalization (Hopper & Traugott, 2003), although this may not always be the case (Meyerhoff, 2000). Aronoff et al. (2005) argue that there is relatively greater use of iconic representation strategies in sign languages than in spoken languages, and this allows for the more rapid development of this kind of morphological complexity. I review work on indicating verbs in Auslan and BSL (de Beuzeville et al., 2009; Cormier et al., 2015; Fenlon et al., 2018), on challenges to the notion of agreement in sign languages (Schembri et al., 2018a), difficulties with applying typological comparisons of morphological ‘complexity’ cross-linguistically (Schembri et al., 2018b), and map out future work on sign language morphology as part of the new European Research Council ‘SignMorph’ project. This project will involve investigating the factors that are relevant to an understanding the nature of sign language morphology (1) the age of sign languages and the nature of grammaticalization processes, (2) the role of iconicity in mapping grammatical meanings onto form, and (3) the sociolinguistic typology of signing communities, particularly the highly variable nature of sign language acquisition and transmission.