  In this article we introduce the notion of cubical $(\omega,p)$-categories,
for $p \in \mathbb N \cup \{\omega\}$. We show that the equivalence between
globular and groupoid $\omega$-categories proven by Al-Agl, Brown and Steiner
induces an equivalence between globular and cubical $(\omega,p)$-categories for
all $p \geq 0$. In particular we recover in a more explicit fashion the
equivalence between globular and cubical groupoids proven by Brown and Higgins.
  We also define the notion of $(\omega,p)$-augmented directed complexes, and
show that Steiner's adjunction between augmented directed complexes and
globular $\omega$-categories induces adjunctions between $(\omega,p)$-augmented
directed complexes and both globular and cubical $(\omega,p)$-categories.
  Combinatorially, the difficulty lies in defining the appropriate notion of
invertibility for a cell in a cubical $\omega$-category. We investigate three
such possible definitions and the relationship between them. We show that
cubical $(\omega,1)$-categories have a natural structure of symmetric cubical
categories. We give an explicit description of the notions of lax, oplax and
pseudo transfors between cubical categories, the latter making use of the
notion of invertible cell defined previously.
