The characteristics of common cuckoo (Cuculus canorus) and host eggs are widely thought to 24 have coevolved over time, but few studies have tested this prediction. We compared cuckoo 25 eggs with those of its primary host, the great reed warbler (Acrocephalus arundinaceus) from 26 four time periods spanning >100 years (between 1900 and 2014), and studied if cuckoo 27 eggshell patterns better resembled those of their hosts over time. We used image analysis to 28 compare five eggshell pattern variables, relating to marking size, diversity, contrast, coverage, 29 and distribution on the egg surface. Each feature showed different temporal trends. All but 30 one of these variables ("dispersion" of spots among egg regions) were species-specific and 31 differed between hosts and parasites. The magnitude of change was greater for hosts than 32 2 cuckoos, which could be a consequence of host eggs' more intensive and variable spottiness. 33Specifically, the proportion of the egg surface covered with pattern increased marginally over 34 time, and the dispersion of spotting became more even over the egg surface. Egg marking 35 contrast showed a decreasing trend, with species differences also decreasing, suggesting 36 better mimicry. Our results suggest multidirectional evolution of eggshell components in this 37 system, with potential implications for mimicry and rejection over time. Many hosts have evolved the ability to recognize alien eggs and eject them from their nests, 50 or desert their clutch entirely, and so many female cuckoos deceive host parents through 51 laying eggs that mimic those of their host, making recognition of foreign eggs more difficult. 52The eggshells of many bird species are highly variable in colour and spotting among 53 individuals (Underwood & Sealy, 2002; Cassey et al., 2010b; Spottiswoode & Stevens, 2010; 54 Hauber, 2014), and brood parasitism is one of the most important selective pressures in 55 modifying these phenotypic traits (Øien, Moksnes & Røskaft, 1995; Soler & Møller, 1996; 56 Kilner et al., 2006). The hosts' main anti-parasite defence mechanism, egg rejection 57 behaviour, is a complex process (e.g. Pozgayová et al., 2011; de la Colina et al., 2012), where 58 background coloration plays a key role in foreign egg recognition by hosts (e.g. Avilés, 2008; 59 Cassey et al., 2008). In contrast, the function of spottiness has less frequently been studied 60 compared to colour, but hosts also likely use various characters of spotting in egg recognition, 61 for example in identifying foreign eggs (Moskát et al., 2008a), or looking for signatures 62 indicating own egg identity (Stoddard, Kilner & Town, 2014), especially those encoded at the 63 blunt pole of the eggs (e.g. Polaciková & Grim, 2010 (Medina et al., 2016; Brulez et al., 2016). Previous modelling has predicted that 71 cuckoos should over time benefit from increasing the resemblance of their own eggs to those 72 of the host, and that hosts should be under selection to escape parasitism (Takasu, 2003). 73 Therefore, the ...