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This paper is primarily devoted to a class of interpolation inequalities ofHardy and Rellich types on the Heisenberg group $\mathbb{H}^n$. Consequently,several weighted Hardy type, Heisenberg-Pauli-Weyl uncertainty principle andHardy-Rellich type inqualities are established on $\mathbb{H}^n$. Moreover, newweighted Sobolev type embeddings are derived. Finally, an integral inequalityfor vector fields in a domain of the Heisenberg group is obtained, leading toseveral specific weighted Hardy type inequalities by making careful choices ofvector fields.

Generalised Hardy type and Rellich type inequalities on the Heisenberg group