This is an expository work on cubical homology as developed in [S. Eilenberg and S. Mac~Lane, Amer. J. Math. {\bf 75} (1953), 189--199; MR0052766], together with connections [R. Brown, Indag. Math. (N.S.) {\bf 29} (2018), no.~1, 459--482; MR3739625], viewed as a sort of extra-degeneracies. In a previous paper, R. Brown and P.~J. Higgins proved that over a suitable category, the category of chain complexes and the category of crossed complexes are equivalent [Homology Homotopy Appl. {\bf 5} (2003), no.~1, 49--52; MR1989612]. \par The main result (Theorem 10) in this paper, by defining natural filtrations in each dimension and proceeding inductively, shows that for a cubical complex $K$, the connection chain complex ${\rm Con}(K)$ and all of the subcomplexes ${\rm Con}^{p,0}(K)$, ${\rm Con}^{\infty,q}(K)$, ${\rm Con}^{0,q}(K)$, ${\rm Con}^{q,\infty}(K)$ are acyclic in every dimension $n>0$. \par There is also an appendix on homology of cubical sets and homology of their geometric realization.