The geometry of constant mean curvature surfaces in hyperbolic 3-space depends on the range of mean curvature. In case mean curvature H is greater than 1, then there exist locally bijective correspondences between CMC surfaces in Euclidean 3-space. In addition CMC surfaces with H=1 correspond to minimal surfaces in euclidean 3-space. On the other hand, in case H is less than 1, CMC surfaces have corresponding surfaces in neither in Euclidean 3-space nor in 3-sphere. In this sense, CMC surfaces in hyperbolic 3-space with mean curvature less than 1 are peculiar objects in hyperbolic geometry. In my talk, I would like to report my recent work jointly with Josef F. Dorfmeister(Munich) and Shimpei Kobayashi (Hirosaki) on the loop group method for constructing simply connected CMC surfaces with H less than 1 (including minimal surfaces). The key observation of this method is the equivalence of CMC condition and harmonicity of appropriate Gauss maps.