Many different shape degrees of freedom play crucial roles in determining the nuclear ground state and saddle point properties and the fission path. By breaking both the axial and the spatial reflection symmetries simultaneously, we have developed multidimensionally-constrained covariant density functional theories (MDC-CDFTs) in which all shape degrees of freedom βλμ with even μ, such as β20, β22, β30, β32, β40, etc., are included self-consistently. The MDC-CDFT’s have been applied to the study of fission barriers and potential energy surfaces of actinide nuclei, third minima in potential energy surfaces of light actinides, shapes and potential energy surfaces of superheavy nuclei, the Y32 correlations in N= 150 isotones and Zr isotopes, and shapes of hypernuclei.

In this contribution we introduce MDC-CDFT’s and focus on applications to tetrahedral nuclear shapes. With functionals DD-PC1 and PC-PK1, the ground state shape of 110 Zr is predicted to be tetrahedral so is that of 112 Zr with DD-PC1. The tetrahedral shape originates from large energy gaps around Z= 40 and N= 70 when the β32 distortion is allowed. With the functional DD-PC1, β32>0.03 and the energy gain due to the β32 distortion is larger than 300 keV for the ground states of 248Cf and 250Fm with N= 150.