The rates and patterns of spread of 63 species in the Czech Republic and seven in Britain and Ireland were determined from quadrat records, mapovací pole (11- × 12-km mapping zones in the Czech Republic), and 10- × 10-km hectads in Britain and Ireland. Species that have straight sections on logarithmic, square root, and arithmetic plots were found; the last group (10 of them) are generally casuals. Straight square root plots (52 found) are expected from Fisher-Skellam reaction diffusion models, logarithmic equivalents (36 found) where the dispersal kernel has a thicker tail or where spread involves scattered colonies. The greater frequency of the straight square root plots is unexpected. The straightness of so many plots shows that there is no significant variation in recorder bias. About a third of the species showed lags, and many species showed a later slowing of the rate of spread, called here bends. Lags may indicate an initial casual phase as well as slower growth; bends, a lack of further areas to spread to. Typically, with much variation, areas of occupancy double in about 10 y (logarithmic mode) or spread at about 2 km · y⁻¹ (square root mode). Both speeds indicate that most spread is from human activity. The reasons for the major phenomena (the occurrence of the two types of spread predicted by models, their distinctness, and the distribution in rates) may reflect economic and landscape features as well as biological ones. The straightness, speed, and patterns found are all new results for invasion biology.