Tissue chips (TCs) and microphysiological systems (MPSs) that incorporate human cells are novel platforms to model disease and screen drugs and provide an alternative to traditional animal studies. This review highlights the basic definitions of TCs and MPSs, examines four major organs/tissues, identifies critical parameters for organization and function (tissue organization, blood flow, and physical stresses), reviews current microfluidic approaches to recreate tissues, and discusses current shortcomings and future directions for the development and application of these technologies. The organs emphasized are those involved in the metabolism or excretion of drugs (hepatic and renal systems) and organs sensitive to drug toxicity (cardiovascular system). This article examines the microfluidic/microfabrication approaches for each organ individually and identifies specific examples of TCs. This review will provide an excellent starting point for understanding, designing, and constructing novel TCs for possible integration within MPS.