Background: The objective evaluation of the olfactory function of coronavirus disease 2019 patients is difficult because of logistical and operator-safety problems. For this reason, in the literature, the data obtained from psychophysical tests are few and based on small case series. Methods: A multicenter, cohort study conducted in seven European hospitals between March 22 and August 20, 2020. The Sniffin-Sticks test and the Connecticut Chemosensory Clinical Research Center orthonasal olfaction test were used to objectively evaluate the olfactory function. Results: This study included 774 patients, of these 481 (62.1%) presented olfactory dysfunction (OD): 280 were hyposmic and 201 were anosmic. There was a significant difference between self-reported anosmia/hyposmia and psychophysical test results (p = 0.006). Patients with gastroesophageal disorders reported a significantly higher probability of presenting hyposmia (OR 1.86; p = 0.015) and anosmia (OR 2.425; p < 0.001). Fever, chest pain, and phlegm significantly increased the likelihood of having hyposmia but not anosmia or an olfactory disturbance. In contrast, patients with dyspnea, dysphonia, and severe-to-critical COVID-19 were significantly more likely to have no anosmia, while these symptoms had no effect on the risk of developing hyposmia or an OD. Conclusions: Psychophysical assessment represents a significantly more accurate assessment tool for olfactory function than patient self-reported clinical outcomes. Olfactory disturbances appear to be largely independent from the epidemiological and clinical characteristics of the patients. The non-association with rhinitis symptoms and the high prevalence as a presenting symptom make olfactory disturbances an important symptom in the differential diagnosis between COVID-19 and common flu.