Iterated pressure-correction projection schemes for the unsteady incompressible Navier-Stokes equations are developed, analyzed and exemplified, in relation to preconditioned iterative methods and the pressure-Schur complement equation. Typical pressure-correction schemes perform only one iteration per stage or time step, and suffer from splitting errors that result in spurious numerical boundary layers and a limited order of convergence in time. We show that performing iterations not only reduces the effects of the splitting errors, but can also be more efficient computationally than merely reducing the time step. We devise stopping criteria to recover the desired order of temporal convergence, and to drive the splitting error below the time-integration error. We also develop and implement the iterated pressure corrections with both multi-step and multi-stage time integration schemes. Finally, to reduce further the computational cost of the iterated approach, we combine it with an Aitken acceleration scheme. Our theoretical results are validated and illustrated by numerical test cases for the Stokes and Navier-Stokes equations, using implicit-explicit (IMEX) backwards differences and Runge-Kutta time-integration solvers. The test cases comprise a now classical manufactured solution in the projection method community and a modified version of a more recently proposed manufactured solution. The different error types, stopping criterion, recovered orders of convergence, and acceleration rates are illustrated, as well as the effects of the rotational corrections and time-integration schemes. It is found that iterated pressure-correction schemes can retrieve the accuracy and temporal convergence order of fully-coupled schemes and are computationally more efficient than classic pressure-correction schemes.