  We study the Cauchy problem for general, nonlinear, strictly hyperbolic
systems of partial differential equations in one space variable. First, we
re-visit the construction of the solution to the Riemann problem and introduce
the notion of a nondegenerate (ND) system. This is the optimal condition
guaranteeing, as we show it, that the Riemann problem can be solved with
finitely many waves, only; we establish that the ND condition is generic in the
sense of Baire (for the Whitney topology), so that any system can be approached
by a ND system. Second, we introduce the concept of inner speed variation and
we derive new interaction estimates on wave speeds. Third, we design a wave
front tracking scheme and establish its strong convergence to the entropy
solution of the Cauchy problem; this provides a new existence proof as well as
an approximation algorithm. As an application, we investigate the
time-regularity of the graph solutions $(X,U)$ introduced by the second author,
and propose a geometric version of our scheme; in turn, the spatial component
$X$ of a graph solution can be chosen to be continuous in both time and space,
while its component $U$ is continuous in space and has bounded variation in
time.
