Genus in topology

Can somebody provide a formal definition of genus in topology? I find it difficult to imagine what genus is. For example, objects of genus zero are the ones that homeomorphic to a sphere? What about higher genus?

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The classification theorem of closed surfaces tells us that any connected closed surface is homeomorphic to one of:

$1)$ The unit sphere

$2)$ The connected sum of $g$ tori (surface of genus $g$)

$3)$ The connected sum of $k$ real projective planes.

(Where $k,g$ are positive integers.)

So if a particular surface falls into the second category (that is to say, is homeomorphic to a connected sum of $g$ tori) we say that it has genus $g$. It can be pictured as the number of "holes" in the surface.

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Accepted (although it sounds circular). There are many classifications theorems on WP. Which one is it? –  saadtaame Feb 27 '13 at 14:16
@saadtaame "The classification theorem of closed surfaces" is what I know it as, if that doesn't come up with anything try "the classification of closed $2$-dimensional manifolds". (since that is what a closed surface really is!) –  Tom Oldfield Feb 27 '13 at 14:18