We study canonical filtrations of finite-dimensional associative and Lie algebras over the complex numbers, which correspond to optimal destabilizing one-parameter subgroups in the sense of geometric invariant theory (GIT). We establish some fundamental properties of these filtrations, and show that an algebra is semisimple if and only if it is GIT semistable. We give a method to compute the canonical filtration of algebras whose automorphism group contains a sufficiently large torus, and use it to compute these filtrations in some examples.