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Many cephalopod species grow throughout their lifetime. Critically, this means that they lack an asymptotic phase of growth, when, for a substantial part of the lifetime, growth slows and body size increases minimally. Understanding the form of the growth curve requires an understanding of the growth processes operating at several biological levels including the relative growth of organs, muscle fibre production and growth, and at the level of proximal composition and protein synthesis. There are key differences in growth processes between fish and cephalopods; cephalopods have a sac-like body form that provides greater surface area for respiration, continuous production of new muscle fibres that ensures a supply of somatic material for growth, and high retention of synthesised protein. These characteristics provide process-orientated explanations for non-asymptotic growth in cephalopods. However, differences found in growth curves of laboratory-reared animals (two-phase growth curve) and of wild animals (single growth curve) suggests that future work will be needed to resolve this paradox. We need to determine the generality of growth processes observed to date, and how biotic and abiotic factors modify these processes during the lifetime of the animals.