The report builds upon the paper "Conformal Prediction Beyond Exchangeability" by R.F. Barber et al. and aims to explore the impact of lifting the usual assumptions of exchangeability and symmetry on both split and full conformal inference methods. Additionally, throughout the report we conduct a constant comparison of split and full conformal inference methods, taking into account the fact that there is a trade off between computational expensiveness/feasibility and statistical efficiency (split-full trade off). After explaining the split and full conformal inference methods through theory and application, we unravel what happens when only the exchangeability assumption is lifted but symmetry still holds. Subsequently, we analyze the setting where both assumptions are violated and its theoretical repercussions on the coverage bounds. We then discuss a hybrid method -Jackknife+- that addresses the split-full trade off. Finally, we show empirical applications on simulated data points and a real data set to convey the main findings of the paper.