Abstract: For simplicity, let $k$ be a field and $R$ be the power series ring $k[[T_1,\ldots,T_n]]$. In this talk, we discuss a systematic and robust way of approximating finite type schemes over $R$ using schemes essentially of finite type over $k$, preserving various types of singularities and homological properties. This allows us to extend known results and constructions for varieties to finite type schemes over $R$, including existence of rational curves, deformation of singularities, big Cohen-Macaulay algebras, and results in étale cohomology. If time permits, I will discuss other settings this approximation idea can be applied in. This is joint work in progress with Shizhang Li and Bogdan Zavyalov.