A partition of m is said to be unitary if the sum of the reciprocals of the parts is unity. If each m, B < m < or = 2B + 9, has a unitary partition, then so does each larger m. The easy problem is to find the smallest B. If the parts are required to be distinct, the problems are distinctly harder. Are there infinitely many m's with no unitary partition into distinct parts? If not, let D be the largest (i.e. the analogue of the smallest B). Graham showed that D < or = 77 and D. H. Lehmer (unpublished) showed that 77 has no unitary partition into distinct parts. Graham's paper has an impressive list of partitions for 77 < m < or = 156, and for odd m, 156 < m < or = 333. We take a fresh look at these problems.