The USGA and R&A announced a series of proposed changes to the Rules of Golf, which would take effect in 2019. Here's a complete list of the topics addressed, including what the new rule would be and what the current rule is.
BALL AT REST: Player accidentally moves his or her ball during search
New rule: No penalty.
Current rule: 1-stroke penalty.

Player accidentally moves his or her ball or ball-marker on the putting green
New rule: No penalty.
Current rule: 1-stroke penalty (with exceptions).

Standard for deciding whether the player caused his or her ball to move
New rule: The player will be found to be the cause only when it is known or virtually certain (meaning at least 95%) to be the case.
Current rule: Weight of the evidence/more likely than not.

How to replace a ball lying off the putting green when it moves and its exact original spot isn’t known
New rule: Replace the ball on its estimated spot; if that spot was on, under or against attached natural objects, replace the ball on that spot on, under or against those objects.
Current rule: Drop the ball as near as possible to the estimated spot.

BALL IN MOTION: Player’s ball in motion accidentally hits the player, his or her caddie, the person attending the flagstick or the attended or removed flagstick
New rule: No penalty.
Current rule: 1-stroke penalty (expect it is a 2-stroke penalty when the accidental deflection relates to the flagstick or the attendant).

TAKING RELIEF: Where a ball must be dropped
New rule: Drop in a defined relief area.
Current rule: Sometimes the drop is in a specified area, sometimes it is on or as near as possible to a spot or a line.

Where a dropped ball must come to rest
New rule: The ball must come to rest in the relief area where it was dropped, or else the ball must be re-dropped.
Current rule: The ball must be re-dropped if it rolls to any of the nine specified areas (Rule 20-2c), such as rolling more than 2 club-lengths from where the dropped ball struck the ground.

Measuring the size of the relief area where a ball must be dropped and played
New rule: The relief area is measured by a fixed distance of 20 inches or 80 inches from the reference point or the reference line; this can readily be measured by using markings on the shaft of a club.
Current rule: Measured by using 1 or 2 club-lengths (with any length club the player chooses).

How to drop a ball
New rule: The only requirement is for the player to hold the ball above the ground without it touching any growing thing or other natural or artificial object, and let it go so that it falls through the air before coming to rest; to avoid any doubt, it is recommended that the ball be dropped from at least one inch above the ground or any growing thing or object.
Current rule: Stand erect, hold the ball at shoulder height and arm’s length.

Time allowed for a ball search
New rule: A ball is lost if not found in three minutes.
Current rule: A ball is lost if not found in five minutes.

Substituting ball
New rule: A player may always substitute a ball when taking relief.
Current rule: The player must use the original ball when taking free relief (with exceptions); a substituted ball is allowed only when taking penalty relief.

Embedded ball
New rule: A player may take relief without penalty for an embedded ball anywhere (except in sand) in the “general area” (new term for “through the green”), unless a Local Rule has been adopted restricting relief only to areas cut to fairway height or less.
Current rule: Relief is allowed only in areas cut to fairway height or less, unless a Local Rule has been adopted allowing relief anywhere (except in sand) through the green.

AREAS OF THE COURSE:
Replacing a ball on the putting green when it moves from its spot after it already had been lifted and replaced
New rule: The ball must always be replaced on its original spot, even if it was blown by the wind or moved for no clear reason.
Current rule: The ball is replaced only if a player or outside agency caused it to move; otherwise, the ball is played from its new location.

Repairing damage on the putting green
New rule: A player may repair almost any damage (including spike marks and animal damage but not including natural imperfections) on the putting green.
Current rule: A player may only repair ball-marks or old hole plugs on the putting green.

Player touches the line of putt or touches the putting green in pointing out a target
New rule: No penalty, so long as doing so does not improve the conditions for the player’s stroke.
Current rule: Loss of hole/2-stroke penalty (with exceptions).

Putting with an unattended flagstick left in the hole
New rule: No penalty if a ball played from the putting green (or anywhere else) hits the unattended flagstick in hole.
Current rule: Loss of hole/2-stroke penalty if the ball is played from the putting green and hits the unattended flagstick in hole.

Areas the Committee may mark as a penalty area (where relief with 1-stroke penalty is allowed)
New rule: Red- and yellow-marked “penalty areas” may now cover areas of desert, jungle, lava rock, etc., in addition to areas of water.
Current rule: Relief is allowed only from “water hazards.”

Player moves loose impediments, touches the ground with hand or club or grounds the club in a penalty area when the ball is in the penalty area
New rule: No penalty.
Current rule: Loss of hole/2-stroke penalty (with exceptions).

Expanded use of red-marked penalty areas
New rule: Committees are given the discretion to mark all penalty areas as red so that lateral relief is always allowed (but they may still mark penalty areas as yellow where they consider it appropriate).
Current rule: All water hazards should be marked yellow, except where their location on the course makes it impossible or unreasonable to drop behind the hazard; only when this is the case may these water hazards be marked red as lateral water hazards.

Elimination of the opposite side relief option for red penalty areas

New rule: A player is no longer allowed to take relief from a red penalty area on the opposite side from where the ball last entered that penalty area, unless the Committee adopts a Local Rule allowing it.

Current rule: A player is always allowed to take relief from the opposite side of a red-marked lateral water hazard.

Player moves or touches a loose impediment in a bunker when the ball is in the bunker
New rule: No penalty.
Current rule: Loss of hole/2-stroke penalty (with exceptions).

Player touches sand in a bunker with his or her hand or a club when the ball is in the bunker
New rule: No penalty except when a player touches sand (1) with his or her hand or club to test the conditions of the bunker or (2) with the club in the area right behind or in front of the ball, in making a practice swing or in making the backswing for the stroke.
Current rule: Any touching of sand with hand or club results in loss of hole/2-stroke penalty (with exceptions).

Unplayable ball relief options
New rule: A player may take relief outside the bunker back on a line from the hole through where ball was at rest for 2 penalty strokes.
Current rule: No relief outside the bunker (other than in playing from where the player’s last stroke was made).

EQUIPMENT:
Use of clubs damaged during round
New rule: A player may keep using any damaged club, even if the player damaged it in anger.
Current rule: A player may use the damaged club only if it was damaged in the “normal course of play.”

Adding clubs to replace a club damaged during round
New rule: A player may not replace a damaged club, unless the player was not responsible for the damage.
Current rule: A player may replace a damaged club if it is “unfit for play” and was damaged in the “normal course of play.”

Use of distance-measuring devices (DMDs)
New rule: The use of DMDs is allowed, unless a Local Rule has been adopted prohibiting their use.
Current rule: DMD use is prohibited, unless a Local Rule has been adopted allowing their use.

PLAYING A BALL: Caddie standing behind a player to help with that player’s alignment
New rule: A caddie is not allowed to stand on a line behind a player while the player is taking his or her stance and until stroke is made.
Current rule: A caddie is allowed to stand on a line behind a player while the player is taking a stance and preparing to play, but must not stand there while the player makes the stroke.

Caddie lifts and replaces the player’s ball on the putting green
New rule: A caddie may lift and replace the player’s ball on the putting green without the player’s specific authorization to do so.
Current rule: 1-stroke penalty if done without the player’s specific authorization.

WHEN TO PLAY DURING ROUND: Recommendations on how to play promptly
New rule: Recommends that players make each stroke in no more than 40 seconds, and usually in less time.
Current rule: No recommendations are given.

Playing out of turn in stroke play
New rule: No penalty (as today), and “ready golf” is encouraged when it can be done in a safe and responsible way.

Current rule: No penalty, but the current Rule is written in a way that may imply that playing out of turn is wrong or is not allowed.

New alternative form of stroke play
New rule: A new “Maximum Score” form of stroke play is recognized, where a player’s score for a hole is capped at a maximum score (such as double par or triple bogey) that is set by the Committee.

Current rule: In standard individual stroke play, players must hole out at every hole; the only recognized alternative forms of stroke play where holing out is not required are Stableford, Par and Bogey.

Eliminating announcement requirements before lifting a ball under certain Rules
New rule: When a player has good reason to mark and lift a ball to identify it, check for damage or see if it lies in a condition where relief is allowed (such as to see whether it is embedded), the player is no longer required first to announce to another player or his or her marker the intent to do so or to give that person an opportunity to observe the process.

Current rule: Before lifting in these cases, the player must announce to another player or the marker that he or she is doing so and allow that person to observe the process.

Player’s reasonable judgment in estimating and measuring under a Rule
New rule: When estimating or measuring a spot, point, line, area or distance under a Rule, a player’s reasonable judgment will not be second-guessed based on later evidence (such as video review) if the player did all that could be reasonably expected under the circumstances to make an accurate estimation or measurement.
Current rule: A player’s judgment is given no particular weight or deference; the Committee decides any issue about the accuracy of the estimation or measurement based on a review of all facts.