  The existence of a uniform upper bound for the maximum number of limit cycles
of planar piecewise linear differential systems with two zones separated by a
straight line has been subject of interest of hundreds of papers. After more
than 30 years of investigation since Lum-Chua's work, it has remained an open
question whether this uniform upper bound exists or not. Here, we give a
positive answer for this question by establishing the existence of a natural
number $L^*\leq 8$ for which any planar piecewise linear differential system
with two zones separated by a straight line has no more than $L^*$ limit
cycles. The proof is obtained by combining a newly developed integral
characterization of Poincar\'{e} half-maps for linear differential systems with
an extension of Khovanski\u{\i}'s theory for investigating the number of
intersection points between smooth curves and a particular kind of orbits of
vector fields.
