We introduce metric temporal description logics (mTDLs) as combinations of the classical description logic ALC with (a) LTL^bin, an extension of the temporal logic LTL with succinctly represented intervals, and (b) metric temporal logic MTL, extending LTL^bin with capabilities to quantitatively reason about time delays. Our main contributions are algorithms and tight complexity bounds for the satisfiability problem in these mTDLs: For mTDLs based on (fragments of) LTL^bin, we establish complexity bounds ranging from EXPTIME to 2EXPSPACE. For mTDLs based on (fragments of) MTL interpreted over the naturals, we establish complexity bounds ranging from EXPSPACE to 2EXPSPACE.