  We extend the theory of complete minimal surfaces in $\mathbb{R}^3$ of finite
total curvature to the wider class of elliptic special Weingarten surfaces of
finite total curvature; in particular, we extend the seminal works of L. Jorge
and W. Meeks and R. Schoen.
  Specifically, we extend the Jorge-Meeks formula relating the total curvature
and the topology of the surface and we use it to classify planes as the only
elliptic special Weingarten surfaces whose total curvature is less than $4
\pi$. Moreover, we show that a complete (connected), embedded outside a compact
set, elliptic special Weingarten surface of minimal type in $\mathbb{R}^3$ of
finite total curvature and two ends is rotationally symmetric; in particular,
it must be one of the rotational special catenoids described by R. Sa Earp and
E. Toubiana. This answers in the positive a question posed in 1993 by R. Sa
Earp. We also prove that the special catenoids are the only connected non-flat
special Weingarten surfaces whose total curvature is less than $8 \pi$.
