We report a new family of Cc-based heavy-fermion compounds whose electronic specific heat coefficients range from about 400 to over 700 mJ/mol-Ce K-2. Crystals in this family form as CenTmIn3n+2m, where T = Rh or Ir, n = 1 or 2, and m = 1, with a tetragonal structure that can be viewed as n-layers of CeIn3 units stacked sequentially along the c-axis with intervening m-layers of TIn2. Ambient and high-pressure studies show that the quasi-2D layers Of CeIn3 produce unconventional superconducting and magnetic ground states. This family should enable new understanding of the relationship between magnetism and superconductivity in heavy-fermion materials and more generally of why heavy-fermion superconductivity prefers to develop in one structure type and not another. (C) 2001 Elsevier Science B.V. All rights reserved.