Searches for scalar top and bottom quarks have been performed with data collected by the ALEPH detector at LEP. The data sample consists of 21.7 pb−1 taken at , and 5.7 pb−1 taken at . No evidence for scalar top quarks or scalar bottom quarks was found in the channels → cχ, → bℓ, and → bχ. For the channel → cχ a limit of 67 GeV/c2 has been set on the scalar top quark mass, independent of the mixing angle between the supersymmetric partners of the left and right-handed states of the top quark. This limit assumes a mass difference between the and the χ of at least 10 GeV/c2. For the channel → bℓ the mixing-angle independent scalar top limit is 70 GeV/c2, assuming a mass difference between the and the of at least 10 GeV/c2. For the channel → bχ, a limit of 73 GeV/c2 has been set on the mass of the supersymmetric partner of the left-handed state of the bottom quark. This limit is valid if the mass difference between the and the χ is at least 10 GeV/c2.