In this talk, I'll discuss the Iwasawa theory of symmetric powers of modular forms, specifically focussing on the phenomenon of L-invariants of their p-adic L-functions. These p-adic invariants arise when there's a glitch in the interpolation property defining the p-adic L-function. The derivative of the p-adic L-function is then conjectured to interpolate the classical L-values up to some factor, called the L-invariant. I'll talk about some ongoing joint work with Andrei Jorza, as well as some past work with Antonio Lei, aimed at computing these L-invariants for all symmetric powers, as well as in the Hilbert modular case. But first, I'll give an introduction to the subject, including some basic examples and motivation!