  We propose categories of $1$-dimensional and multi-dimensional quantum walks.
In the categories, an object is a quantum walk, and a morphism is an
intertwining operator between two quantum walks. The new framework enables us
to discuss quantum walks in a unified way. The purposes of this paper are the
following: (1) We reinterpret known results in our new framework. (2) We show
several new theorems. For example, it is proved that every space-homogeneous
time-periodic analytic quantum walk on $\mathbb{Z}^d$ has a limit distribution
of velocity for every initial unit vector. Analyticity is a very weak
condition. (3) We ask whether there exists a continuous-time quantum walk
$(V^{(t)})_{t \in \mathbb{R}}$ which realizes a given discrete-time quantum
walk $U$. Existence of $(V^{(t)})_{t \in \mathbb{R}}$ is equivalent to that of
a $1$-parameter group of automorphisms $(V^{(t)})_{t \in \mathbb{R}}$ from the
object $U$ to $U$.
