Abstract Using a one-dimensional αω-dynamo model appropriate to galaxies, we study the possibility of dynamo action driven by a stochastic α-effect and shear. To determine the field evolution, one needs to examine a large number of different realizations of the stochastic component of α. The net growth or decay of the field depends not only on the dynamo parameters but also on the particular realization, the correlation time of the stochastic α compared to turbulent diffusion time-scale and the time over which the system is evolved. For dynamos where both a coherent and fluctuating α are present, the stochasticity of α can help alleviate catastrophic dynamo quenching, even in the absence of helicity fluxes. One can obtain final field strengths up to a fraction ∼0.01 of the equipartition field Beq for dynamo numbers |D| ∼ 40, while fields comparable to Beq require much larger degree of α-fluctuations or shear. This type of dynamo may be particularly useful for amplifying fields in the central regions of disc galaxies.