  We introduce a theoretical framework for the new concept of three-dimensional
(3D) twistronics by developing a generalized Bloch band theory for 3D layered
systems with a constant twist angle $\theta$ between successive layers. Our
theory employs a nonsymmorphic symmetry that enables a precise definition of an
effective out-of-plane crystal momentum, and also captures the in-plane moir\'e
pattern formed between neighboring twisted layers. To demonstrate the novel
topological physics that can be achieved through 3D twistronics, we present two
examples. In the first example of chiral twisted graphite, Weyl nodes arise
because of inversion-symmetry breaking, with $\theta$-tuned transitions between
type-I and type-II Weyl fermions, as well as magic angles at which the in-plane
velocity vanishes. In the second example of twisted Weyl semimetal, the twist
in the lattice structure induces a chiral gauge field
$\boldsymbol{\mathcal{A}}$ that has a vortex-antivortex lattice configuration.
Line modes bound to the vortex cores of the $\boldsymbol{\mathcal{A}}$ field
give rise to 3D Weyl physics in the moir\'e scale. We also discuss possible
experimental realizations of 3D twistronics.
