The Banach-Saks property is an important tool in analysis with applications ranging from partial differential equations (PDEs) to calculus of variations and probability theory. We survey the Banach-Saks property for $L^p$-spaces, with a particular emphasis on the case where $p=1$. In other words, we revisit the celebrated result by W. Szlenk (1965) in a more general context, demonstrating that $L^1$-spaces possess the weak Banach-Saks property.