Abstract: In this work, we present efficient numerical schemes based on the Lagrange multiplier approach for the Navier-Stokes equations. By introducing a dynamic equation (involving the kinetic energy, the Lagrange multiplier, and a regularization parameter), we reformulate the original equations into an equivalent system that incorporates the energy evolution process. First- and second-order dynamically regularized Lagrange multiplier (DRLM) schemes are derived based on the backward differentiation formulas and shown to be unconditionally energy stable with respect to the original variables. The proposed schemes require only the solutions of two linear Stokes systems and a scalar quadratic equation at each time step. Moreover, with the introduction of the regularization parameter, the Lagrange multiplier can be uniquely determined from the quadratic equation, even with large time step sizes, without affecting the accuracy and stability of the numerical solutions. Various numerical experiments including the Taylor-Green vortex problem, lid-driven cavity flow, and Kelvin-Helmholtz instability are carried out to demonstrate the performance of the DRLM schemes. Extension of the DRLM method to the Cahn-Hilliard-Navier-Stokes system will also be discussed.