A significant number of countries do not have any river, most of these are desert countries (like the United Arab Emirates and Saudi Arabia) and mini states (like Malta and Monaco).
Still it is quite impressive how such massive countries like Saudi Arabia (2,149,690 km2) have not rivers and still manage to get enough water for there entire population (33,000,000 people).
Today about 50% of drinking water in Saudi Arabia comes from desalination, 40% from the mining of non-renewable groundwater and only 10% from surface water in the mountainous southwest of the country. The government stores the water underground with the help of aquifers.
Full list of countries without rivers:
Bahamas
Bahrain
Comoros
Kiribati
Kuwait
Maldives
Malta
Marshall Islands
Monaco
Nauru
Oman
Qatar
Saudi Arabia
Tuvalu
United Arab Emirates
Vatican City
Yemen