  This note is based on a series of lectures delivered in Kyoto University.
This note surveys the homogeneous Besov space $\dot{B}^s_{pq}$ on ${\mathbb
R}^n$ with $1 \le p,q \le \infty$ and $s \in {\mathbb R}$ in a rather
self-contained manner. Possible extensions of this type of function spaces are
breifly discussed in the end of this article. In particular, the fundamental
properties are stated for the spaces $\dot{B}^s_{pq}$ with $0<p,q \le \infty$
and $s \in {\mathbb R}$ and $\dot{F}^s_{pq}$ with $0<p<\infty$, $0<q \le
\infty$ and $s \in {\mathbb R}$ as well as nonhomogeneous coupterparts
$B^s_{pq}$ with $0<p,q \le \infty$ and $s \in {\mathbb R}$ and $F^s_{pq}$ with
$0<p<\infty$, $0<q \le \infty$ and $s \in {\mathbb R}$.
