  This work proposes a unified theory of regularity in one hypercomplex
variable: the theory of $T$-regular functions. In the special case of
quaternion-valued functions of one quaternionic variable, this unified theory
comprises Fueter-regular functions, slice-regular functions and a
recently-discovered function class. In the special case of Clifford-valued
functions of one paravector variable, it encompasses monogenic functions,
slice-monogenic functions, generalized partial-slice monogenic functions, and a
variety of function classes not yet considered in literature. For $T$-regular
functions over an associative $*$-algebra, this work provides integral
formulas, series expansions, an Identity Principle, a Maximum Modulus Principle
and a Representation Formula. It also proves some foundational results about
$T$-regular functions over an alternative but nonassociative $*$-algebra, such
as the real algebra of octonions.
