  In an attempt to understand the origins and the nature of the law binding
together two group operations into a {\em skew brace}, introduced in [L.\
Guarnieri \& L.\ Vendramin, Math.\ Comp.\ \textbf{86} (2017), 2519--2534] as a
non-Abelian version of the {\em brace distributive law} of [W.\ Rump, J.\
Algebra {\bf 307} (2007), 153--170] and [F.\ Ced\'o, E.\ Jespers \& J.\
Okni\'nski, Commun.\ Math.\ Phys.\ {\bf 327} (2014), 101--116], the notion of a
{\em skew truss} is proposed. A skew truss consists of a set with a group
operation and a semigroup operation connected by a modified distributive law
that interpolates between that of a ring and a brace. It is shown that a
particular action and a cocycle characteristic of skew braces are already
present in a skew truss; in fact the interpolating function is a 1-cocycle, the
bijecitivity of which indicates the existence of an operation that turns a
truss into a brace. Furthermore, if the group structure in a two-sided truss is
Abelian, then there is an associated ring -- another feature characteristic of
a two-sided brace. To characterise a morphism of trusses, a {\em pith} is
defined as a particular subset of the domain consisting of subsets termed {\em
chambers}, which contains the kernel of the morphism as a group homomorphism.
In the case of both rings and braces piths coincide with kernels. In general
the pith of a morphism is a sub-semigroup of the domain and, if additional
properties are satisfied, a pith is a $\mathbb{N}_+$-graded semigroup. Finally,
giving heed to [I.\ Angiono, C.\ Galindo \& L.\ Vendramin, Proc.\ Amer.\ Math.\
Soc.\ {\bf 145} (2017), 1981--1995] we linearise trusses and thus define {\em
Hopf trusses} and study their properties, from which, in parallel to the
set-theoretic case, some properties of Hopf braces are shown to follow.
