The binding of quarks into hadrons is one of our time's most difficult theoretical physics problems. It is a quantum, relativistic, highly non-linear system that cannot be simplified through perturbative expansions. Yet, such binding is responsible for the constitution of nuclear matter, and its theoretical understanding is critical for fundamental physics. A method of growing importance in solving this problem is the use of lattice field theory simulations, which allow for a direct, ab initio numerical evaluation of correlation functions in the relevant field theories, quantum chromodynamics (QCD) and quantum electrodynamics (QED). Lattice simulations are statistical system simulations with billions of degrees of freedom, and existing supercomputing technologies directly bound their predictive capacity. In November 2023, the Frontier supercomputer at Oak Ridge National Laboratory passed the Exaflop barrier (10^18 operations per second). Although such a barrier is essentially artificial, it generally drives large technological investments and deep transformations in computational sciences. As we stand at the frontier between the Petascale and Exascale Eras, I will review the achievements of the Petascale Era, which saw the emergence of high-precision hadronic physics using lattice QCD, from the perspective of my work. From this, I will extrapolate which breakthrough one can expect in hadronic and nuclear theory for the Exascale Era.