Explore the theoretical foundations of numerical relativity in this comprehensive lecture by Helvi Witek from the University of Illinois. Delve into the 3+1 decomposition of spacetime and Einstein's equations, examine the well-posedness of hyperbolic equations, and discover well-posed formulations of Einstein's equations. Learn about suitable coordinate choices for stable numerical simulations and gain insights into the applications of numerical relativity. Cover key topics including the decomposition of spacetime, propagation of points, projection operators, and the decomposition of Einstein's equations. Prepare for a hands-on xTensor tutorial by downloading the provided Mathematica notebook and installing the free xAct/xTensor package. This lecture serves as an essential introduction to the field, complementing recommended textbooks for in-depth study of Numerical Relativity.