Symbolic dynamics provides simple yet surprisingly rich models for dynamical systems, though many fundamental classification problems remain unresolved. A central example is the conjugacy problem for subshifts of finite type (SFTs), whose complexity is reflected in deep links to combinatorics and matrix theory. Building on the seminal work of Cuntz and Krieger, one can associate graph C*-algebras and Leavitt path algebras to SFTs in a way that recovers classical invariants while also producing new (operator) algebraic obstructions to conjugacy.   In this talk, I will present recent advances in understanding the relationship between symbolic dynamics and graph algebras, including a resolution of a question of Eilers showing that eventual conjugacy of SFTs coincides with graded homotopy equivalence of the associated graph C*-algebras. These results illustrate how graph algebras, in both the analytic and purely algebraic settings, not only shed light on longstanding problems in dynamics but also open new avenues at the interface of functional analysis, pure algebra, non-commutative geometry, and quantum mathematics.