  In this review paper, we survey Hardy type inequalities from the point of
view of Folland and Stein's homogeneous groups. Particular attention is paid to
Hardy type inequalities on stratified groups which give a special class of
homogeneous groups. In this environment, the theory of Hardy type inequalities
becomes intricately intertwined with the properties of sub-Laplacians and more
general subelliptic partial differential equations. Particularly, we discuss
the Badiale-Tarantello conjecture and a conjecture on the geometric Hardy
inequality in a half-space of the Heisenberg group with a sharp constant.
