Enteroviruses are associated with various diseases accompanied by rare but severe complications. In recent years, outbreaks of enterovirus D68 and enterovirus A71 associated with severe respiratory infections and neurological complications have been reported worldwide. Since information on molecular epidemiology in respiratory samples is still limited, the genetic diversity of enteroviruses was retrospectively analysed over a 4-year period (2013–2016) in respiratory samples from paediatric patients. Partial viral major capsid protein gene (VP1) sequences were determined for genotyping. Enteroviruses were detected in 255 (6.1%) of 4187 specimens. Phylogenetic analyses of 233 (91.4%) strains revealed 25 different genotypes distributed to Enterovirus A (39.1%), Enterovirus B (34.3%), and Enterovirus D (26.6%). The most frequently detected genotypes were enterovirus D68 (26.6%), coxsackievirus A6 (15.9%), and enterovirus A71 (7.3%). Enterovirus D68 detections were associated with lower respiratory tract infections and increased oxygen demand. Meningitis/encephalitis and other neurological symptoms were related to enterovirus A71, while coxsackievirus A6 was associated with upper respiratory diseases. Prematurity turned out as a potential risk factor for increased oxygen demand during enterovirus infections. The detailed analysis of epidemiological and clinical data contributes to the non-polio enterovirus surveillance in Europe and showed high and rapidly changing genetic diversity of circulating enteroviruses, including different enterovirus D68 variants.