A rev iew on selected areas of researc h on low -di mensiona l str uctures consistin g of diluted magnetic (semimagnetic ) semiconductors is attem pted . I n the intro ductor y part , af ter short historica l prelimi nari es, w e descri be some early ideas and speculati ons concerning prop erties of such structures and w e conf ront them w ith real results obtained in the course of their exp erimental investigations. Several issues are addressed to but, in view of the richness of the Ùeld, the review is by no means not pretending to be an exhaustiv e one. W e begin w ith early studies of metal {insul ator{s emico ndu ctor structures and progress to structures grow n by molecula r b eam epitaxy . W e discuss, in particular, the idea of a spin sup erlattice, typ e-I-to-typ e-I I transition, wave functions mapping, and detailed determinati on of the proÙles of realisti c quantum w ells. W e put special emphasis on digital grow th and, speciÙcall y, parab oli c and half -parab oli c quantum w ells, use of tunable g -factor in studies of trions, self-assembled quantum dots, spin inj ection and, Ùnally, on hybrid structures of ferromagnets and diluted magnetic semiconductor structures. J . Ko ssut the term appl ies to semi conducti ng sol id sol uti ons i n whi ch one of cati on com p onents i s substi tuted by a m agneti cal ly acti ve species. Ori gi nal l y, the na m e deno ted m ostl y I I{ VI comp ounds wi th a fra cti on of cati ons substi tuted by m anganese. La ter, the numb er of comp ounds i ncl uded i nto the DMS fam ily ha s expanded and no w i t encom pa ssesa wi de num b er of comp ounds : II{ VIs, I I I{ Vs, IV{ VIs, I I{ Vs, etc. , wi th m any m agneti c cati ons i ntro duced i nto thei r respecti ve l atti ces: Mn, Co , Cr, V, Fe, Eu, etc. Som e very earl y work has b een do ne i n W arsaw and i ts descri pti on can be found i n the revi ews quoted ab ove. Ini ti al moti vati on for the earl y i nterest wa s vari ed: i t ori gi na ted pri m ari l y in an i ntenti on to study an i nteracti on of magneti c and \ semi conducti ng " pro p erti es o f materi al s, but i t stem med al so f rom an extra ordi nari l y hi g h el ectro n m obi li ty observed i n Hg Mn T e by Mo rri ssy [6] and then, m ore speciÙcal ly, fro m, has l ater b ecome kno wn as the gi ant spi n spl i tti ng , observed i n CdMnT e [7,8] and ZnMnT e [9]. Apa rt fro m these m oti vati ons, i n the course of i nvesti gatio ns underta ken, there app eared a spectrum of i nteresti ng pheno mena tha t singl ed out di luted magneti c semi conducto rs as a di sti nct and uni que gro up of m ateri al s. The most i nteresti ng feature of the physi cs of DMS ari ses, of course, fro m the intera cti on b etween conducti on a nd/ or val ence el ectro ns wi th the m agneti c com p onent vi a the spin-dep endent s p À d i ntera cti on. The l i tera ture devoted to these m ateri al s is vast and covers m any subj ects ra ngi ng fro m thei r m agneti c pro p erti es, l atti ce pro p erti es, electro n tra nsport pro p erti es, and opti cal and magneto ...