  This work is concerned with the derivation of a robust a posteriori error
estimator for a discontinuous Galerkin method discretisation of linear
non-stationary convection-diffusion initial/boundary value problems and with
the implementation of a corresponding adaptive algorithm. More specifically, we
derive a posteriori bounds for the error in the $L^2(H^1)$-type norm for an
interior penalty discontinuous Galerkin (dG) discretisation in space and a
backward Euler discretisation in time. An important feature of the estimator is
robustness with respect to the P\'{e}clet number of the problem which is
verified in practice by a series of numerical experiments. Finally, an adaptive
algorithm is proposed utilising the error estimator. Optimal rate of
convergence of the adaptive algorithm is observed in a number of test problems.
