In　this　work,　we　investigate　the　epitaxial　growth　of　InAs　quantum　dots　(QDs)　on　Ge　substrates.　By　varying　the　growth　parameters　of　growth　temperature,　deposition　thickness　and　the　growth　rate　of　InAs,　high　density　(1.2　x　10(11)　cm(-2))　self-assembled　InAs　QDs　were　successfully　epitaxially　grown　on　Ge　substrates　by　solid-source　molecular　beam　epitaxy　and　capped　by　Ge　layers.　Pyramid-　and　polyhedral-shaped　InAs　QDs　embedded　in　Ge　matrices　were　revealed,　which　are　distinct　from　the　lens-　or　truncated　pyramid-shaped　dots　in　InAs/GaAs　or　InAs/Si　systems.　Moreover,　with　a　200　nm　Ge　capping　layer,　one-third　of　the　embedded　QDs　are　found　with　elliptical　and　hexagonal　nanovoids　with　sizes　of　7-9　nm,　which,　to　the　best　of　our　knowledge,　is　observed　for　the　first　time　for　InAs　QDs　embedded　in　a　Ge　matrix.　These　results　provide　a　new　possibility　of　integrating　InAs　QD　devices　on　group-IV　platforms　for　Si　photonics.