Make slip knot and knit first stitch on. For each new stitch after that, insert the right needle between 2 stitches; wrap yarn as for knitting(A). Draw through a new stitch and transfer it to left needle(B).

Method 1—Emily Ocker’s Circular Beginning
First, you are going to make a ring with the required number of stitches, only these are going to be crocheted.
1. Make a loop. Do not pull it tight; keep it loose. Grasp the loop where the strands meet with your left thumb and index finger.

You can make the loop with the tail going up. You will then have to weave in the tail.
2. Insert the hook into the loop from front to back. Grab the yarn with the hook (yo) and bring through the loop. You now have one loop on the hoop.
3. Insert the hook into the loop again; you will be going under the yarn tail as well. Hook the yarn (yo) and bring through the loop. Hook the yarn again and bring through the loop on the hook as if you were making a chain. You now have two loops on the hook. Repeat this step until you have the required number of loops.
4. Carefully replace the hook with your needles and begin knitting in the round. After you have finished a few rounds, pull the tail to close up the circle.
You now have a nice, tight circle with no hole in the middle and the end worked in.
—updated January 7, 2017

Crocheters! Check here for a similar method to begin crocheting in the round.

Method 2
Cast on the required number of stitches and divide evenly between two needles, not the usual 3 or 4 needles. Using a third needle, knit in the round for a few rounds. If the pattern includes increases, when you have 12 or more stitches, it will probably be comfortable for you to now introduce another needle. Think "Magic Loop," only with double pointed needles.
—added April 23, 2011 (first posted on sheepthrillsyarncompany.com January 2011)

Method 3
Make a slip knot, leaving a 12" tail. The knot should be snug, not tight. Hold the knot in the left hand and the needle in the right hand. *Yarn over (1 st created); insert the needle into the slip knot’s loop, wrap yarn around needle and pull through the loop (another st created). Rep from * until you have the desired number of stitches. After you have knitted 3 or 4 rounds, pull the tail of the slip knot to close up the hole.
—revised March 19, 2013 from "Knitting in the Round" by the House of White Birches, pg. 162

• Make a slip knot and place it on a crochet hook the same size as the needles. (Figure 1)
• With yarn in back of the knitting needle, bring the hook over the top of the needle (Figure 2), catch the yarn and pull it through the loop on the hook. (Figure 3) Repeat this step until the number of stitches desired has been cast on.

• Make a slip knot with a 6" tail and put it on the bottom needle.
• Wrap the working yarn around the needles three times by going to the back of the top needle and then over the top and towards you (Figure 1).
• With working yarn, knit the three stitches on the top needle.
• Rotate the needles so the slip st is on the top needle.
• Remove slip knot and undo so that you can knit with the yarn tail. With both yarn tail and working yarn, knit 3 stitches on each needle. You now have 12 stitches. (Figure 2)
• With working yarn, knit one round. In Figure 3, you can see the way the stitches flow over the toe.
• Next round: *K1, kfb (knit in front and back of stitch); repeat from * around (18 stitches).
  Optional: Use a lifted increase instead: K1, right-slanting lifted increase in next stitch; knit that stitch
• Knit one round.
• Next round: *K2, kfb (knit in front and back of stitch); repeat from * around (24 stitches).
  Optional: Place a marker after every increase.
• Repeat the last 2 rounds increasing 6 stitches each round until you have the desired number of stitches. Move to 3 or 4 needles when you have enough stitches to knit comfortably.
  —revised August 28, 2012

• Step 1: Knit the next st and leave on needle.
• Step 2: Insert the tip of the left needle into the new st and slide to left needle.

• Often, we run out of the “tail” when casting on a large number of stitches. To avoid this, use one end of the skein as your working yarn and the other end (or another ball of yarn) as the tail and cast on. Your yarn tail will never be too short again!
  —added January 17, 2012, Esther Bozak, Twisted Stitches
  
• The tail of the yarn, when using a long-tail cast on, should be about 4 times the width of your finished edge.
  —added March 16, 2013

Picot cast on

• Using a cable cast-on or other knit cast-on, cast on 4 stitches.
• Now we are going to begin a 5-stitch picot. Cast on 5 more stitches. Bind off 5 stitches. (Try the modified conventional bind-off; once learned, it is faster.) Slip the stitch from the right- to the lefthand needle by inserting the lefthand needle in the stitch from front to back. You now have 4 stitches.
• *Cast on 9 stitches (4 stitches plus the 5 stitches for the picot). Bind off 5 stitches. Slip the stitch from the right- to the lefthand needle by inserting the lefthand needle in the stitch from front to back. You have just added 4 stitches to the total cast on.*
• Repeat between * * until you have 4 stitches less than the total desired.
• Cast on 4 stitches.
  —added November 8, 2011 from Lucy Neatby’s Knitting Gems

• Using a smooth yarn in a contrasting color, make a slip knot and place it on a crochet hook the same size as the needles. (Figure 1)
• With yarn in back of the knitting needle, bring the hook over the top of the needle (Figure 2), catch the yarn and pull it through the loop on the hook. (Figure 3) Repeat this step until the number of stitches desired has been cast on.
  Step 3: Chain a few stitches. Cut the yarn and pull through the last chained stitch. When you are ready to unravel the waste yarn, this will be the end that you gently pull to remove the waste yarn.
• Fiigure 4 shows the completed cast on.
  —added December 1, 2011 from Lucy Neatby’s Knitting Gems 4