Spiders in the world which have been linked to fatalities in humans, or have been shown to have potentially fatal bites by toxicology studies of their venom, include:
The Brazilian wandering spider
The Australasian funnel-web spider
The widow spiders, including the Australian redback spider
The six-eyed sand spider, and possibly other spiders of genus Sicarius
The recluse spiders
Spiders which likely are not deadly to humans, but which are nonetheless medically significant include:


The hobo spider (this is doubtful, however)
The yellow sac spider
Certain species of tarantulas
The false black widows
Spiders which can inflict painful bites (often similar to a bee sting), but whose bites generally do not cause any systemic or long-lasting effects, include:

The huntsman spider
The redback jumping spider (not to be confused with the very dangerous redback spider, the latter being one of the widow spiders).