Excitons and charged excitons (trions) are investigated in ZnSe-based quantum well structures with (Zn,Be,Mg)Se and (Zn,Mg)(S,Se) barriers by means of magneto-optical spectroscopy. Binding energies of negatively-(X − ) and positively (X + ) charged excitons are measured as functions of quantum well width, free carrier density and in external magnetic fields up to 47 T. The binding energy of X − shows a strong increase from 1.4 to 8.9 meV with decreasing quantum well width from 190 to 29Å. The binding energies of X + are about 25% smaller than the X − binding energy in the same structures. The magnetic field behavior of X − and X + binding energies differ qualitatively. With growing magnetic field strength, X − increases its binding energy by 35-150%, while for X + it decreases by 25%. Zeeman spin splittings and oscillator strengths of excitons and trions are measured and discussed.