The number of alveoli is a key structural determinant of lung architecture. A design-based stereologic approach was used for the direct and unbiased estimation of alveolar number in the human lung. The principle is based on two-dimensional topology in three-dimensional space and is free of assumptions on the shape, size, or spatial orientation of alveoli. Alveolar number is estimated by counting their openings at the level of the free septal edges, where they form a two-dimensional network. Mathematically, the Euler number of this network is estimated using physical disectors at a light microscopic level. In six adult human lungs, the mean alveolar number was 480 million (range: 274–790 million; coefficient of variation: 37%). Alveolar number was closely related to total lung volume, with larger lungs having considerably more alveoli. The mean size of a single alveolus was rather constant with 4.2 × 10⁶μm³ (range: 3.3–4.8 × 10⁶μm³; coefficient of variation: 10%), irrespective of the lung size. One cubic millimeter lung parenchyma would then contain around 170 alveoli. The method proved to be very efficient and easy to apply in practice. Future applications will show this approach to be an important addition to design-based stereologic methods for the quantitative analysis of lung structure.