Enteroviruses (EVs) are associated with a wide spectrum of diseases involving various organs. Our aim was to give a historical overview of the genesis of clinical sample processing for EVs in the Slovak Republic (SR) during the 1958–2020 period, within the framework of the World Health Organization (WHO) polio program. Further, analyses were made of the data obtained from the archives of processed clinical sample surveillance using statistical methods. We used generalized additive models (GAM) with binomial distribution and logit link functions and an autoregressive moving average (ARMA) to analyze the data obtained during this 63-year period. Our results show trends in the composition of EV strains circulating in the population. Furthermore, statistically significant increasing trends of the non-polio enteroviruses (NPEVs) were observed over the studied time, represented by echoviruses (E) and coxsackieviruses A and B (CVA and CVB), with a cyclical pattern of occurrence. The most prevalent serotype over this period was CVB5, which became significantly more prevalent after 2000. While PVs, CVB1, and CVB3 were present in the second half of the studied period, CVA10, CVA16, E3, E25, and E30 appeared more frequently.