In our talk we will discuss the old and celebrated question regarding the pointwise behavior of Fourier Series near $L^1$. This presentation will include \begin{itemize} \item the resolution of Konyagin's conjecture (ICM, Madrid 2006) on the pointwise convergence of Fourier Series along lacunary subsequences; \item the $L^1$-strong convergence of Fourier Series along lacunary subsequences. \item recent progress on the $L^1$-strong convergence of (the full) Fourier Series. \end{itemize} We end with several considerations on the relevance/impact of the above items on the subject of the pointwise convergence of Fourier Series.