We start with the definition of braid group as the fundamental group of configuration space and as the mapping class group of a disc with punctures. Braids admit generalizations in various directions. The are also special types of braids defined among all braids by specific properties. One of natural  generalizations of braids is the monoid of partial braids. We interprete it as a monoid of certain isotopy classes of homeomorphisms of punctured disc. Brunnian braid is a braid that becomes trivial after removing any one of its strands. We consider Brunnian braids on surfaces. In the cases of sphere and projective plane Brunnian braids are connected with homotopy groups of 2-dimensional sphere. Cohen braids are related to Brunnian braids and we study their properties also.