If the answer is no, have you tried houseruling it in, and what effect did it have on your game?
10 yes (sort of), 20 no.
Taking 20 was not a thing in 4th edition, and is similarly absent from 5th. Taking 10 however is sort of a thing in 5th, but it's not implemented the way you might think it should be.
Basically, the way "taking 10" works in 5e is that every ability (And by extension, skill), has a "passive" score (Basic Rules v2 page 59). This is 10 + modifier, and it sort of represents your natural, not at all under pressure ability in a specific discipline.
The best example of this is Passive Perception. Basically, if you walk into a room, your passive perception is what you instantly notice. Many items in 5e have a higher DC if someone is passively looking for it than when they are actively doing so. But Passive Perception doesn't draw attention that actively looking might.
Ultimately, when it comes to taking 20, this gets back to a fundamental D&D principle. If failure isn't interesting on a specific roll, there is no sense in rolling the check at all. This is the problem that take 20 solved, and while 5e could fall victim to it, I've found in practice that it really doesn't. I've run several sessions where there was no need to roll dice at all, as situations pretty much just were RP and the characters often found automatic success on things like Diplomacy or bluff etc because their passive scores are high, but also because the DCs were really low for things like that (what can I say, folks in Phandalin aren't shy when it comes to telling PCs stuff).
Since the release of the DMG, yes
The 5e DMG was released three months after this question and the currently accepted answer by wax eagle were originally posted. In it is the following passage, under "Multiple Ability Checks" (DMG, page 237 - emphasis mine):
Sometimes a character fails an ability check and wants to try again. In some cases, a character is free to do so; the only real cost is the time it takes. With enough attempts and enough time, a character should eventually succeed at the task. To speed things up, assume that a character spending ten times the normal amount of time needed to complete a task automatically succeeds at that task. However, no amount of repeating the check allows a character to turn an impossible task into a successful one.
Thus, if a character has any chance of succeeding at a check, the DM estimates how long one check would take, multiplies that by ten, and declares the character to be successful after that amount of time.
Since "any chance of success" implies that a character would succeed on a roll of 20, 5e is essentially allowing the results of a 3e/3.5e Take 20 action but with only ten times the amount of time of a regular check (rather than twenty times the regular amount).
The other main difference is that the 3.5e rules on "taking 10" and "taking 20" explicitly say that Take 10 and Take 20 cannot be done when a character is "threatened or distracted". The 5e DMG rule does not make this explicit, but it is implied in "the only real cost is the time it takes" (i.e., there is no cost of prolonged exposure to threat and distraction).
D&D Next, as of present, has no direct equivalent to either Take-10 nor Take-20. At present, 4 "final" rulebooks have been released: The Starter Set, the Players Basic Rules, the DM's Basic Rules, and the Player's Handbook. Yet to be released are the Dungeon Master's Guide, and the Monster Manual.
The passive scores, generated using a base of 10, are similar in concept to Take-10, but are used primarily in situations where no active effort is used. Players are not given the option to "take 10" on a called for roll, with the exception of a few specific class abilities.
There are certain class abilities which replace die rolls with minimums. For example, the Rogue's Reliable Talent replaces rolls of ≤9 on proficient abilitiy checks with a minimum 10. This is conceptually not the same as the take 10, tho' the effect in play is not too dissimilar.