Introduction to Sweep Picking

As far as modern lead guitar techniques go, sweep picking is the one that inspires the most awe and fear. Listeners, viewers, and live audiences are spellbound by the sheer velocity at which masters of the technique are able to fire off arpeggios and scale fragments. 

Now while the sweep picking technique is undoubtedly impressive, it’s also often the most difficult to master. However, once you master the technique, the possibilities become limitless.
The sweep picking concept is to play a rapid succession of notes, using one fluid up or down movement of the picking hand across the strings – without letting any notes ring into one another like a chord.

the picking hand

Often overlooked, the picking hand is, in my opinion, far more important and requires far more attention than the fretting hand, especially at the early stages of learning the technique. 
Most beginners start by playing one string at a time, using one short upward or downward movement for each string. This common error can result in guitarists never learning to sweep quickly. Fortunately, this is simple to remedy.
The easiest way to stop playing a succession of short up or down strokes, one string at a time, is to aim for the last string in the direction you are sweeping.
In the example below (a very simple A minor arpeggio), you’d aim for the E string when sweeping downwards (ascending the arpeggio), and for the G string when sweeping upwards (descending the arpeggio). The idea is to sweep right through all three strings in one movement, without pausing. 
You can make this motion made much more fluid by concentrating on another simple action. Make sure you’re not sweeping with your wrist. Instead, keep your wrist stiff and sweep with your whole forearm, bending at the elbow. 
By doing this, you’ll only have to lean one position with your picking hand, then simply move it up or down by bending your elbow. Picking with a wrist action is great for fast and accurate alternate and directional picking, but getting comfortable with sweeping from the elbow early on will help greatly when learning sweep-tapping and 5, 6, and 7 string sweeps later on. 
Using the elbow and keeping the wrist in one position also helps mute the adjacent strings that have just been played. The picking hand muting is done with either the outer edge of the palm, or the thumb. 

the fretting hand

The fretting hand positions are fairly basic when fretting one string with one finger: the fingers just follow each other up or down the strings. All you need to concentrate on is lifting your finger off the previous string just as you play the next string.
The tricky part comes when you need to fret two or more adjacent strings with one finger. This action is called rolling. The aim here is to roll your finger from one string to the other, simultaneously fretting one string and muting the other.
In the example below, start by fretting the note C (5th fret) on the G string with the tip of your finger. Then roll your finger so that the tip lifts and mutes the G string, while the area halfway between the tip and the knuckle is fretting the B string. 
Now roll your finger a little more so that the tip and the area between the tip and the knuckle are muting the G and B strings respectively, and use the area closest to the knuckle to fret the E string. To sweep back up (descending the arpeggio), simply reverse the process.
Make sure that you mute the previous string just as you play the next one, ensuring that the notes don’t ring into one another. 
For an interesting way to apply the sweep picking technique, check out my introduction to voice leading, covering combining major and minor arpeggios.