A process for reducing the content of oxygen-containing and/or nitrogen-containing compounds in streams having an isobutene content of at least 10% by weight, which comprises passing the stream in the liquid state at a temperature T [in K] and a linear velocity v [in cm/min] over a fixed bed of an acid-free zeolite having a mean pore size of from 0.3 to 1.5 nm, where the fixed bed has a length l [in cm] in the flow direction of the stream and T, v and l obey the relationship 2.sup.(T-283 K)/10 Kl/v.ltoreq.500 min, is described. The process avoids the formation of isobutene oligomers.