In this talk I will explain new research on $L$-invariants of modular forms, including ongoing joint work with Robert Pollack. $L$-invariants, which are $p$-adic invariants of modular forms, were discovered in the 1980’s, by Mazur, Tate, and Teitelbaum. They were formulating a $p$-adic analogue of Birch and Swinnerton-Dyer’s conjecture on elliptic curves. In the decades since, $L$-invariants have shown up in a ton of places: $p$-adic $L$-series for higher weight modular forms or higher rank automorphic forms, the Banach space representation theory of $\mathrm{GL}(2,\mathbb{Q}_p)$, $p$-adic families of modular forms, Coleman integration on the $p$-adic upper half-plane, and Fontaine’s $p$-adic Hodge theory for Galois representations. In this talk I will focus on recent numerical and statistical investigations of these $L$-invariants, which touch on many of the theories just mentioned. I will try to put everything into the context of practical questions in the theory of automorphic forms and Galois representations and explain what the future holds.