This paper reviews the main estimation and prediction results derived in the context of functional time series, when Hilbert and Banach spaces are considered, specially, in the context of autoregressive processes of order one (ARH(1) and ARB(1) processes, for H and B being a Hilbert and Banach space, respectively). Particularly, we pay attention to the estimation and prediction results, and statistical tests, derived in both parametric and non-parametric frameworks. A comparative study between different ARH(1) prediction approaches is developed in the simulation study undertaken. A review and comparative study on functional time series techniques