Viscoelastic flows have been a central subject for the modelling of complex flows, first because of the potential direct applications to many non-Newtonian industrial flows (i.e. to numerous polymer suspensions) but also because conceptually, it is an ideal multi-physics framework to unify the mechanics of (viscous) fluids and (elastic) solids, and subsequently develop a huge number of applications with fluid/solid interactions. Among many closure formulas used to define the non-Newtonian (Cauchy) stress in the momentum balance equations (ie in the so-called Navier-Stokes equations, within the usual Eulerian description of large-deformation flows), the Upper-Convected Maxwell (UCM) equations are widespread. When complemented by (purely viscous) Newtonian stresses, the system of equations defines so-called Oldroyd-B fluids, a paradigmatic viscoelastic model. Oldroyd-B fluids have continuously been the focus of numerical studies for decades. Indeed, in the regime where viscoelastic stresses are almost purely elastic, numerical simulations of Oldroyd-B fluids hardly converge [AOP]. In this talk, we will report about the current status and perspectives of the non-standard approach we initiated in [Boy21] to solve the UCM equations: include them in a symmetric-hyperbolic quasi-linear system of PDEs consistent with polyconvex elastodynamics. The latter formulation ensures well-posedness of the Cauchy initial-value problem on small times, for compressible viscoelastic flows where deviatoric stresses satisfy UCM equations. It allows introducing viscoelasticity while starting from established discretizations of elastodynamics [Lee et al.]. REFERENCES [AOP] M. Alves, P. Oliveira and F. Pinho, Numerical Methods for Viscoelastic Fluid Flows, Annual Review of Fluid Mechanics, Annual Reviews, 2021, 53, 509-541 [Boy21] S. Boyaval, Viscoelastic flows of Maxwell fluids with conservation laws, ESAIM Math. Model. Numer. Anal., 2021, 55, 807-831 [Lee et al.] C.H. Lee, A.J. Gil, T. Jaugelavicius, T. Richardson, S. Boyaval, D. Violeau, J. Bonet, Symmetrisation and hyperbolicity of first-order conservation laws in large strain compressible viscoelasticity using the Smoothed Particle Hydrodynamics method, CMAME, in press