Enhancement of the quality of laser wake-field accelerated (LWFA) electron beams implies the improvement and controllability of the properties of the wake waves generated by ultra-short pulse lasers in underdense plasmas. In this work we present a compendium of useful formulas giving relations between the laser and plasma target parameters allowing one to obtain basic dependences, e.g. the energy scaling of the electrons accelerated by the wake field excited in inhomogeneous media including multi-stage LWFA accelerators. Consideration of the effects of using the chirped laser pulse driver allows us to find the regimes where the chirp enhances the wake field amplitude. We present an analysis of the three-dimensional effects on the electron beam loading and on the unlimited LWFA acceleration in inhomogeneous plasmas. Using the conditions of electron trapping to the wake-field acceleration phase we analyse the multi-equal stage and multiuneven stage LWFA configurations. In the first configuration the energy of fast electrons is a linear function of the number of stages, and in the second case, the accelerated electron energy grows exponentially with the number of stages. The results of the two-dimensional particle-in-cell simulations presented here show the high quality electron acceleration in the triple stage injection–acceleration configuration.