We show that the morphisms from the braid group with n strands in the mapping class group of a surface with a possible non empty boundary, assuming that its genus is smaller or equal to n/2 are either cyclic morphisms (their images are cyclic groups), or transvections of monodromy morphisms (up to multiplication by an element in the centralizer of the image, the image of a standard generator of the braid group is a Dehn twist, and the images of two consecutive standard generators are two Dehn twists along two curves intersecting in one point). As a corollary, we determine the endomorphisms, the injective endomorphisms, the automorphisms and the outer automorphism group of the following groups: the braid group with n strands where n is greater than or equal to 6, and the mapping class group of any surface of genus greater or equal than 2. For each statement involving the mapping class group, we study both cases: when the boundary is fixed pointwise, and when each boundary component is fixed setwise. We will also describe the set of morphisms between two different braid groups whose number of strands differ by at most one, and the set of all morphisms between mapping class groups of surfaces (possibly with boundary) whose genus (greater than or equal to 2) differ by at most one.