  We introduce a new phase field model for tumour growth where viscoelastic
effects are taken into account. The model is derived from basic thermodynamical
principles and consists of a convected Cahn--Hilliard equation with source
terms for the tumour cells and a convected reaction-diffusion equation with
boundary supply for the nutrient. Chemotactic terms, which are essential for
the invasive behaviour of tumours, are taken into account. The model is
completed by a viscoelastic system constisting of the Navier--Stokes equation
for the hydrodynamic quantities, and a general constitutive equation with
stress relaxation for the left Cauchy--Green tensor associated with the elastic
part of the total mechanical response of the viscoelastic material.
  For a specific choice of the elastic energy density and with an additional
dissipative term accounting for stress diffusion, we prove existence of
global-in-time weak solutions of the viscoelastic model for tumour growth in
two space dimensions $d=2$ by the passage to the limit in a fully-discrete
finite element scheme where a CFL condition, i.e. $\Delta t\leq Ch^2$, is
required.
  Moreover, in arbitrary dimensions $d\in\{2,3\}$, we show stability and
existence of solutions for the fully-discrete finite element scheme, where
positive definiteness of the discrete Cauchy--Green tensor is proved with a
regularization technique that was first introduced by J. W. Barrett, S. Boyaval
("Existence and approximation of a (regularized) Oldroyd-B model". In: M3AS.
21.9 (2011), pp. 1783--1837). After that, we improve the regularity results in
arbitrary dimensions $d\in\{2,3\}$ and in two dimensions $d=2$, where a CFL
condition is required. Then, for $d=2$, we pass to the limit in the
discretization parameters and show that subsequences of discrete solutions
converge to a global-in-time weak solution.
  Finally, we present numerical results.
