Density functional theory (DFT) at the hybrid B3LYP level has been applied to Ge-10(z) germanium clusters (z = - 6, - 4, - 2, 0, + 2, + 4, + 6) starting from 12 different initial configurations. The D-4d 4,4-bicapped square antiprism found experimentally in B10H102- and other 10-vertex clusters with 22 skeletal electrons is calculated for the isoelectronic Ge-10(2-) to be the global minimum by more than 15 kcal/mol. The global minima found for electron-rich clusters Ge-10(4-) and Ge-10(6-) are not those known experimentally. However, experimentally known structures for nido-B10H14 and the pentagonal antiprism of arachno-Pd@ Bi-10(4+) are found at higher but potentially accessible energies for Ge-10(4-) and Ge-10(6-). The global minimum for Ge-10 is the C-3v 3,4,4,4-tetracapped trigonal prism predicted by the Wade-Mingos rules and found experimentally in isoelectronic Ni@Ga-10(10-). However, only slightly above this global minimum for Ge-10 ( + 3.3 kcal/mol) is the likewise C-3v isocloso 10-vertex deltahedron found in metallaboranes such as (eta(6)-arene) RuB9H9 derivatives. Structures found for more electron-poor clusters Ge-10(2+) and Ge-10(4+) include various capped octahedra and pentagonal bipyramids. This study predicts a number of 10-vertex cluster structures that have not yet been realized experimentally but would be interesting targets for future synthetic 10-vertex cluster chemistry using vertex units isolobal with the germanium vertices used in this work.