Functions of bounded mean oscillation ($BMO$-functions) were introduced by John and Niremberg in 1961, as a subset of the space $L^1_{\text{loc }}(\mathbb R^n)$ and they applied them to smoothness problems in partial differential equations. In this course we shall restrict ourselves to the one-dimensional case, more precisely to functions defined on the unit circle $\mathbb T$. \par To start with, this class of functions had no relation with complex variables. However, in the middle of the decade of the 1960's, Spanne and Stein, independently, proved that there is a close connection between $BMO$-functions and the conjugation operator. Namely, they proved the following result:

If, $f\in L_{\mathbb R}\sp\infty (\mathbb T )$, then the conjugate function $\tilde f$ has bounded mean oscillation on $\mathbb T $.} \par In other words, they proved a substitute for $L^\infty $ of M. Riesz's conjugation theorem: \par {\it The conjugation operator \, $f\mapsto \tilde f$ \, is bounded from $L\sp\infty _\mathbb R(\mathbb T)$\, to the space $BMO$}.

Later on C.~Fefferman proved that the relation between $BMO$-functions and the conjugation operator is

much more intimate. In fact, he showed that if $f\in L_{\Bbb R}\sp 1(\mathbb T )$ then

$$f\in BMO \,\,\, \iff \,\,\, f=f_1+\tilde {f_2},\quad\hbox{with $f_1, f_2\in L_{\Bbb R}\sp\infty (\mathbb T).$}$$

This result is essentially equivalent to saying that the dual of $H\sp 1$ is $BMOA$, the space of those $f\in H^1$ whose boundary values have bounded mean oscillation in $\mathbb T$.

The interest of complex analysts in the subject comes not only because of the duality theorem but also because it is possible to define $BMO$ in a good number of different ways which connect it with a lot of distinct topics in complex analysis and operator theory. In this course we shall explore some of these connections.