The peculiarity of a dusty plasma is related to the unique capacity of dust particles to self-organize themselves into various structures. Dust structures in a plasma are perfect examples of complex systems because of many various constituents, with intensive interactions between them, as well as various space and time scales involved. The emergence of structures involves numerous processes of generation and loss of plasma electrons and ions, in particular on the surfaces of the dust grains. The main interest is related to the processes of formation, arrangement, symmetry and transitions appearing in open dissipative systems of interacting dust particles. Furthermore, the grain structures are often non-extensive containing a limited number of particles interacting via non-collective long-range forces. The non-extensive structures in complex plasma systems, such as dust particle molecules and clusters, attract particular attention since they can be easily obtained in experiments and allow theoretical treatment at the most fundamental level. Here, attention is paid mainly to the recently observed and/or theoretically fundamental questions.