Canada will eliminate its penny. The reason is cost: Each penny costs 1.6 cents to make. Quoting “Canada's penny withdrawal: All you need to know” on CBC News:

The government says it costs about \$11 million a year to supply pennies to the economy.
How would you pay sums in cash where the cents amount to _y with y not being either 0 or 5? You take the sum of all purchased goods, add taxes and round to the next multiple of 5:
• x1, x2: round down to x0.
• x3, x4: round up to x5.
• x6, x7: round down to x5.
• x8, x9: round up to (x+1)0.
If products only cost multiples of 5 cents then there would never be rounding. But the fact that taxes are applied afterwards is a problem (in contrast to, say, Germany, where it is already contained in prices). One could, however, have prices that become multiples of 5 cents after taxes.

The U.S. would equally profit from eliminating the penny. But it’s not likely that that will happen. Quoting “Canada is dropping its penny – will the U.S. follow suit?” on USATODAY.com:

The U.S. Treasury Department cited a statement from Treasurer Rosie Rios from earlier this year when asked about the Canadian decision. She said the Obama administration has looked at possibly using cheaper materials to make the penny, which is now made of zinc.

That's quite a difference from when candidate Obama was campaigning to become president.

“We have been trying to eliminate the penny for quite some time – it always comes back,” Obama said at the time. “I need to find out who is lobbying to keep the penny.”

Two separate bills calling for the demise of the penny, introduced in 2002 and 2006 by Republican congressman Jim Kolbe, failed to advance in the U.S. House of Representatives.

Other countries have previously successfully abandoned low-value coins. Quoting “Backgrounder: Withdrawing the Penny from Circulation”:
• Australia: removed its one-cent and two-cent coins from circulation in 1992.
• Brazil: stopped issuing one-real coins in 2005.
• Finland: does not issue one-cent and two-cent euro coins since the euro was introduced in 2002.
• Israel: stopped issuing the one-agora coin in 1991 and the five-agorot coin in 2008.
• Netherlands: stopped issuing one-cent and two-cent euro coins in 2004.
• New Zealand: removed its one-cent and two-cent coins from circulation in 1989 and its five-cent coin in 2006.
• Norway: removed its one-øre and two-øre coins in 1972; by 1991, it had also removed its five, ten and twenty-five øre coins.
• Sweden: removed its one-öre and two-öre coins in 1971; by 1992, it had also removed its five, ten and twenty-five öre coins. In 2009, it repealed the fifty-öre coins from circulation.
• Switzerland: officially withdrew its one-centime coin from circulation in 2006, while the two-centime coin lost its legal tender status in 1978.
• United Kingdom: removed the legal tender status of the half-penny in 1984.