We consider implicit/explicit schemes for the numerical solution of the Navier-Stokes equations and develop the stability and optimal error estimates for the schemes. Particularly, we show that the schemes are almost unconditionally stable (with the time step size satisfying τ≤C for some constant C) and convergent for smooth (H2) initial data, weakly unconditionally stable (with τ|log h| ≤C where h is the mesh size) and convergent for non-smooth (H1) initial data, and conditionally stable (with τ ≤ Ch2) for non-smooth (L2) initial data. We also study the Euler implicit/explicit scheme based on the mixed finite element or the so-called time-space finite element iterative method (the T-S method) for the stationary Navier-Stokes equations and establish the almost unconditionally stability and optimal error estimates uniform in time for the scheme.