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Numerous linear expansion cosmological models are critically revised and compared with standard cosmologies from a historical perspective. There are two major classes of such models: those strictly linear along the expansion history (coasting) and those linear at late times, but not in the early Universe (quasi-linear). Their merits and weaknesses, theoretical foundations and compatibility with diverse observational constraints are assessed. The ways the different models, a number of them avoiding inflation, try to solve the cosmological problems of Big Bang cosmology are also considered. Some severe tensions, model-dependent analyses, inconclusive data and unsettled controversy after two decades since the $\varLambda $CDM model became the standard cosmology are pointed out. All in all, at least quasi-linear models cannot be definitely ruled out so far. Certainly, $\varLambda $CDM continues dominating the landscape, but the growing evidence here reported advises paying some attention to the (quasi-)linear expansion scenario as an alternative cosmology.