A fundamental result in algebraic geometry is that the stack Coh$^{P}(X)$ of pure sheaves with Hilbert polynomial $P$ on a projective scheme $X$ has a stratification by Harder-Narasimhan(HN) types. Every pure sheaf possesses a canonical filtration, the Harder-Narasimhan filtration, which determines its HN type. Furthermore, the distinguished stratum of semistable sheaves admits a moduli space. In this dissertation, we study similar instability stratifications and moduli spaces for algebraic stacks over Coh$^{P}(X)$. We introduce a new technique, which we call infinite dimensional geometric invariant theory, which allows us to construct instability stratifications for some stacks of this form, such as stacks of pairs and stacks of $\Lambda$-modules. We also study some problems related to the moduli of pure coherent algebras on a projective scheme $X$. We consider an instability stratification of the stack of finite dimensional algebras, which corresponds to the case when $X$ is a point, and study canonical filtrations of these algebras. We also study the positivity of some line bundles on the moduli space of pure reduced coherent algebras on $X$, which is isomorphic to the moduli space of equidimensional branchvarieties on $X$. Our positivity results establish that this moduli space is projective, partially answering an open question of Alexeev and Knutson.