  We review fundamental issues arising in the exact solution of the
one-dimensional Hubbard model. We perform a careful analysis of the Lieb-Wu
equations, paying particular attention to so-called `string solutions'. Two
kinds of string solutions occur: $\Lambda$ strings, related to spin degrees of
freedom and $k-\Lambda$ strings, describing spinless bound states of electrons.
Whereas $\Lambda$ strings were thoroughly studied in the literature, less is
known about $k-\Lambda$ strings. We carry out a thorough analytical and
numerical analysis of $k-\Lambda$ strings. We further review two different
approaches to the thermodynamics of the Hubbard model, the Yang-Yang approach
and the quantum transfer matrix approach, respectively. The Yang-Yang approach
is based on strings, the quantum transfer matrix approach is not. We compare
the results of both methods and show that they agree. Finally, we obtain the
dispersion curves of all elementary excitations at zero magnetic field for the
less than half-filled band by considering the zero temperature limit of the
Yang-Yang approach.
