"Vehicle Smoking"
A vehicle smoking out the tail can be one of many different problems,below we will cover them.

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"Blue Smoke Out The Tail Pipe"
This color indicates that your burning oil, reasons you could be burning oil are:

PCV Valve
Clogged pcv valve, it's the valve that comes off the valve cover and plugs back into the intake manifold, if this get's clogged.. and they do, you will get excess crank pressure that will cause the oil to push past the piston rings and out the tail pipe burnt oil goes. Replace at any local auto parts store.

Piston Rings
Worn piston rings can allow oil past them into the combustion chamber and burn out the exhaust. To test this you will need a compression test done & possible a leak down test in case the compression test cant detect it from the excess oil in the cylinder.

Valve Stem Seals
 Valve Stem Seals usually only cause burning at fist start, what happen is the oil seeps back down the valve stem when you turn the vehicle off and drips into the cylinder, then when you fire it up for the first time in the morning or when ever you get that nice puff of bluish smoke and then is seem to reside away. Replacing them is the only solution.

Valve Guides
If there is too much clearance between the valve and the guide oil can get past them and burn in the combustion chamber.Symptoms are just like a bad valve stem steal mentioned above.

Turbo Charger
Turbo chargers can leak from the piston ring on the turbine side and the compressor side where the seal is. If the journal bearing get's worn that also can promote leaking due to the shaft starting to change position allowing a bigger clearance between the shaft and the bearing and then more oil gets to the turbine piston ring and out the tail pipe it goes.

Turbo Drain & Angle
A clogged turbo return line can back up if clogged and cause a ton of oil to burn out the turbine side. Disconnect the line and check for any obstructions that may be prohibiting oil returning.The angle of a turbo drain line is crucial, any kinks or sudden incline will back up the oil all the way to the turbo. Make sure you have a nice clean angle from the turbo to the oil pan with no "U" OR  upward "J" shapes.

Crank Case Pressure
Aside from the pcv valve, if your turbo charged and your burning oil and have tried every method mentioned above then you might need to check your crank case pressure. You can do this by pulling the dip stick while the engines at idle, you shouldn't feel hardly any pressure when you place you thumb or finger over it, if you do you have too much crank case pressure and need to invest in a catch can, whether it be a vacuum state one or a open air one. You can also pull the oil cap and put your hand on it to see how much pressure is there. Too much pressure will cause your turbo to leak and also push oil past the piston rings.

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"White Smoke Out The Tail Pipe"
White Smoke Indicates Your Burning Coolant/Antifreeze,reasons you could be burning Coolant/Antifreeze are:

Blown Head Gasket
A blown head gasket, whether it be corroded or just old and not sealing any more will allow coolant into the hot cylinder and out the exhaust the white smoke goes.Pulling the head,inspecting it for damages,re-decking it & putting a new head gasket on is your only option. A local auto part store can do a compression test for free. Common telltale signs are white foam out the radiator cap, over fill bottle not returning coolant when the vehicle cools down and low coolant levels after you've filled it multiple times. You can also check your spark plug tips, if their too clean then your getting coolant in there. In Some cases the coolant will mix with the oil and that when you see a milk chocolate oil, "signs you have a cracked head,a blown head gasket or even a warped head or block".(Check the bottom of the radiator cap for black soot,that will be a sure sign your getting combustion in the cooling system).

Cracked Cylinder Head
In some cases a cracked cylinder head can occur and the only way to find out is by pulling the head unfortunately, symptoms are just like a blown head gasket. Except the coolant gauge increases a lot faster under load.

Cracked Sleeve
You can see if a sleeve is cracked by sliding a bore scope/endoscope into the spark plug hole and checking for any cracks or unnatural lines. If it's cracked it's a complete engine over haul and unless your rebuilding it yourself and have it good with a machine shop you'll need a new block or sleeve. For the average person, replacing the entire engine is the best decision.

Condensation
In some cases in the morning or cold weather it's perfectly normal to see condensation out the tail pipe and mistake it for white smoke, if your vehicle is running fine with no over heating,check engine lights or misfiring then drive it until something comes up.
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"Black/Grey Smoke"
This Indicates Your Burning Too much Fuel,reasons you could be running rich are:

Fuel Injectors
Leaking fuel injectors will cause you to run rich, remove them and see if there is excess gas running off the ends.

FPR (Fuel Pressure Regulator)
A bad fpr whether it be stuck shut or clogged will increase the fuel pressure at the fuel rail causing the injectors to spray more.

Fuel Return Line
If your return fuel line is clogged it will also increase fuel pressure at the fuel rail with the same symptoms as the bad FPR mentioned above.

MAF Sensor (Mass Air Flow Sensor)
If the Maf malfunctions by getting too dirty,has a hardware issue that causes increased voltage or something breaks inside, you can have rich conditions occur because it will tell the ECU that more air is flowing than really is and mess up the Air/fuel mixture. In most cases you can clean the small sensor inside with "Maf cleaner" at any local auto parts store.

O2 Sensor (Oxygen Sensor)
One of the fairly common problem parts. When a O2 sensor goes bad it does 1of2 things, it either runs way rich or way lean. This device measures oxygen levels in the exhaust with a zirconium coating in the sensor which puts out a small voltage when exposed to depreciating levels of oxygen. Reason the O2 sensors go bad are..damage from a blown head gasket that produced steam,burning oil,old age & anti seize. Your probably saying "anti seize"?!? Yes, anti seize contains sulfur, if the past get on the sensor while applying it to the threads and was not clean off, it will eat the sensor causing it to prematurely malfunction. Bottom line, don't get anti seize on the O2 sensor & if you do, clean it off good before installing it.

Carburetor Choke
A stuck or misadjusted automatic choke, leaking float in the fuel bowl, saturated foam float can all be causes of a rich condition in older vehicles.

Injector Timing/Control
In diesel engines injector timing & control will cause excessive black smoke, much more unusual then a hard acceleration will. This all works off of the position of the roller which touches the cam.