The study of geometric aspect of Special Relativity is all about the geometry of Minkowski spacetime $(M,\langle,\rangle)$, a flat spacetime whose curvature vanishes at everywhere. The (Minkowski/Lorentz) inertial frame is defined based on Einstein's 2 postulates of Special Relativity, which in term of mathematic can be interpreted as a proper orthochronous transformation $(t,x):\ M\longrightarrow\mathbb R^4 $, and its representation matrix $L=(a_{ij})_{0\leq i,j\leq 3}$ therefore is a proper orthochronous matrix, i.e $$L^\top\eta\,L\,=\,\eta\,=\,\text{diag}\,(-1,1,1,1),\ \ |L|=1,\ \ a_{00}\geq 1.$$ These transformation are called proper orthochronous Lorentz transformation.
So, it's clear that the inertial frames and Lorentz transformations take an important rule in the study of geometry of Minkowski spacetime. But, in the study of General Relativity where spacetimes need not to be flat and henceforth thet need not to be vector spaces or affine spaces, I wonder what is the role of inertial frames and Lorentz transformations ?