  The global existence of weak solutions of the incompressible viscoelastic
flows in two spatial dimensions has been a long standing open problem, and it
is studied in this paper. We show the global existence if the initial
deformation gradient is close to the identity matrix in $L^2\cap L^\infty$, and
the initial velocity is small in $L^2$ and bounded in $L^p$, for some $p>2$.
While the assumption on the initial deformation gradient is automatically
satisfied for the classical Oldroyd-B model, the additional assumption on the
initial velocity being bounded in $L^p$ for some $p>2$ may due to techniques we
employed. The smallness assumption on the $L^2$ norm of the initial velocity
is, however, natural for the global well-posedness . One of the key
observations in the paper is that the velocity and the \textquotedblleft
effective viscous flux\textquotedblright $\mathcal{G}$ are sufficiently regular
for positive time. The regularity of $\mathcal{G}$ leads to a new approach for
the pointwise estimate for the deformation gradient without using $L^\infty$
bounds on the velocity gradients in spatial variables.
