There are many important and striking differences between classical complex analysis in one variable and complex analysis in several variables. In this talk, we will illustrate this by discussing just one such difference, the Hartogs extension phenomenon. For example, if $D$ denotes the annular domain in $\mathbb C^n$ consisting of the unit ball $B$ minus the closed ball of radius ½, then any holomorphic function in $D$ extends holomorphically to the whole unit ball $B$ ... provided $n\geq 2$; it is clearly not true when $n=1$. This particular result can be proved by using the Cauchy integral formula, but a proof that works in more general situations leads to a study of the $\bar \partial$ equation.