He certainly claims to be amazing with the ability to climb walls, sense danger and punch people in the face really hard, but is Spider-Man really all that in a bag of scientific potato chips?
The geniuses over at ASAPScience explain that for Peter Parker to even become Spider-Man, the spider that bit him would need to pass on a retrovirus. That retrovirus would then have to genetically modify all of the different cells in his body. Not impossible, but highly improbable.
What about strength? Could Spidey be super-strong? ASAPScience explains the fancy retrovirus we talked about earlier would have to create more Actin and Myosin proteins in the body to increase strength in a human being.
As for walking on walls, it's also theoretically possible, but Peter Parker would have to walk around with uber hairy hands all the time due to the microfibres required to make it work.
When it comes to the magical "Spidey Sense", it's also theoretically plausible, but only if the event Spider-Man is being tipped to is going on directly near him. The video explains this well.
So the science of Spider-Man shows us that it's theoretically possible for a man to get about with the powers of a giant spider, but it's highly improbable. Still, good enough for me.