We give an overview of recent results concerning exciton and carrier tunneling and relaxation in II-VI asymmetric double quantum wells structures with semimagnetic (diluted magnetic) wells or barriers. We undertake the issue of the spin relaxation and injection in the tunneling process and we discuss the dependence of the tunneling efficiency and spin injection on the energy detuning between the initial and final state of the process. The fast tunneling of excitons as whole entities was observed with efficiency strongly dependent on the barrier width, exciton detuning and the relation between detuning and optical phonon energy. In the energy transfer processes an important role of the charged excitons formation is observed. As a result of high sensitivity of energy levels in semimagnetic semiconductor to magnetic field, the thermodynamical redistribution of neutral and charged excitons dependence in magnetic field is observed and must be taken into account in the analysis.1 Introduction Asymmetric double quantum well (ADQW) structure forms an excellent laboratory for studying tunneling and related spin effects. It consists of two quantum wells (QWs) of different depth and/or width coupled through a barrier having width comparable to QW widths. In this system, usually the two lowest excitonic states are separated spatially but the overlap between the wave functions enables tunneling. ADQW structure based on the II-VI compounds is particularly useful in such investigations. It results from the possibility of technological realization of diverse designed structures due to a great number of available materials and their combinations. Simultaneously exciton binding energies in wide-gap II-VI semiconductors, few times larger compared to III-V materials, enable investigations of the influence of excitonic effects on the tunneling process. In addition, due to stronger coupling between excitons and longitudinal optical (LO) phonons in II-VI compounds than in III-V ones, phonon assisted tunneling plays a major role in the energy relaxation process and can be studied.The detuning between initial and final state of the tunneling process proves to be the parameter that controls the tunneling mechanism and, consequently, its efficiency. With the development of crystal growth techniques it became possible to accurately control the structure parameters and, as a consequence, the energy levels and wave function penetration through the barrier. However, to perform a detailed study of tunneling, it is preferable to tune continuously the energy levels and thus change tunneling conditions. A straightforward way of tuning the levels is using a diluted magnetic semiconductor (DMS) as a QW material. Due to s,p-d exchange interaction between manganese ions and band carriers, a giant Zeeman effect leads to 100 times larger band splitting than in a nonmagnetic material [33]. Therefore, a very efficient tuning of energy levels is possible with little modification of single particle or exci-