Shakespeare does everything at once: tells a story, develops characters, writes beautifully, teaches history, and -- oh yeah -- startles us with randy humor. 

Elizabethan England was a raunchier time.  If you were witty, there were no limits. 

One critic says the great dramatist wrote more than 1,100 puns on sex and genitals. 

Shakespeare's bountiful gifts have been a problem for prudes and teachers ever since.  Shakespeare is often taught with a hope and a prayer that students won't get it.  Best example of all: "Much Ado About Nothing."  (Men have something, so to speak, women have "nothing."  I know what you're thinking: OMG.)

Shakespeare's audience didn't need anything explained.  They got it.  (A lot of Shakespeare is like a high-class version of "Who's on First?") 

Assume he wants you to be uproariously entertained, he's working on lots of levels, and if it sounds risqué, it is. 

1. Romeo and Juliet (Act I, Scene 1)

GREGORY: The quarrel is between our masters and us their men.

SAMPSON: 'Tis all one, I will show myself a tyrant: when I have fought with the men, I will be cruel with the maids, and cut off their heads.

GREGORY: The heads of the maids?

SAMPSON: Ay, the heads of the maids, or their maidenheads; take it in what sense thou wilt.

GREGORY: They must take it in sense that feel it.

SAMPSON: Me they shall feel while I am able to stand: and 'tis known I am a pretty piece of flesh.

(Sampson is a military stud, dispatching enemy soldiers and virgins with equal abandon.  While he is able to stand.  Cocky self-esteem has rarely been so well expressed as in "'tis known I am a pretty piece of flesh.")

2. Comedy of Errors (Act III, Scene 2)

ANTIPHOLUS: Then she bears some breadth?

DROMIO: No longer from head to foot than from hip to hip:

she is spherical, like a globe; I could find out countries in her.

ANTIPHOLUS: In what part of her body stands Ireland?

DROMIO: Marry, in her buttocks: I found it out by the bogs.

ANTIPHOLUS: Where Scotland?

DROMIO: I found it by the barrenness; hard in the palm of the hand.

ANTIPHOLUS: Where France?

DROMIO: In her forehead; armed and reverted, making war against her heir.

ANTIPHOLUS: Where England?

DROMIO: I looked for the chalky cliffs, but I could find no whiteness in them; but I guess it stood in her chin, by the salt rheum that ran between France and it.

(Ireland is in her buttocks, out by the bogs.  Shakespeare seems to be saying the Irish live in an unpleasant place.  I'm betting the London audience roared.  "Barrenness hard in the palm of the hand" must be the mons Veneris.  "Making war against her heir" might seem a lightweight pun but surely references the civil war in France still going ten years before play was written.)

3. Hamlet (Act III, Scene 2)

HAMLET: Lady, shall I lie in your lap? [Lying down at OPHELIA's feet]

OPHELIA: No, my lord.

HAMLET: I mean, my head upon your lap?

OPHELIA: Ay, my lord.

HAMLET: Do you think I meant country matters?

OPHELIA: I think nothing, my lord.

HAMLET: That's a fair thought to lie between maids' legs.

OPHELIA: What is, my lord?

HAMLET: Nothing.

(Lap and country matters are definite salacious puns.  Country matters is a well-known term for what peasants and farm animals do in the haystacks.  Country is what it sounds like.  Nothing?  Probably also a pun, which means the opposite of what it seems to say.  Oh, nothing, and as well that other nothing in Much Ado About Nothing.)

4. Macbeth (Act II Scene 3)

MACDUFF: Was it so late, friend, ere you went to bed, that you do lie so late?

PORTER: 'Faith, Sir, we were carousing till the second cock: and drink, Sir, is a great provoker of three things.

MACDUFF: What three things does drink especially provoke?

PORTER: Marry, sir, nose-painting, sleep and urine. Lechery, Sir, it provokes, and unprovokes: it provokes the desire, but it takes away the performance: therefore, much drink may be said to be an equivocator with lechery: it makes him, and it mars him; it sets him on, and it takes him off; it persuades him and disheartens him; makes him stand to, and not stand to: in conclusion, equivocates him in a sleep, and giving him the lie, leaves him.

MACDUFF: I believe drink gave me the lie last night.

(Just a fancy riff on how booze makes a man hot and then not.)

5. Anthony and Cleopatra (Act I, Scene 2)

IRAS: Am I not an inch of fortune better than she?

CHARMIAN: Well, if you were but an inch of fortune better than I, where would you choose it?

IRAS: Not in my husband's nose.

(She wants another inch farther down.)

6. Henry IV / Part I (Act III, Scene 3)

MISTRESS QUICKLY: ... I am an honest man's wife, and, setting thy knighthood aside, thou art a knave to call me so.

FALSTAFF: Setting thy womanhood aside, thou art a beast to say otherwise.

MISTRESS QUICKLY: Say, what beast, thou knave, thou?

FALSTAFF: What beast? Why, an otter!

PRINCE HAL: An otter, Sir John! Why an otter?

FALSTAFF: Why, she's neither fish nor flesh; a man knows not where to have her.

MISTRESS QUICKLY: Thou art an unjust man in saying so: thou or any other man knows where to have me, thou knave, thou!

(This is just ruthless, zany insults flying.  Falstaff wants to provoke her.  He says she's a big blob a man wouldn't know what to do with.  She insists that a man most certainly would.)

7. The Taming of the Shrew (Act II, Scene 1)

PETRUCHIO: Come, come, you wasp; i' faith, you are too angry.

KATHARINA: If I be waspish, best beware my sting.

PETRUCHIO: My remedy is then, to pluck it out.

KATHARINA: Ay, if the fool could find it where it lies.

PETRUCHIO: Who knows not where a wasp does wear his sting? In his tail.

KATHARINA: In his tongue.

PETRUCHIO: Whose tongue?

KATHARINA: Yours, if you talk of tails: and so farewell.

PETRUCHIO: What, with my tongue in your tail? nay, come again, Good Kate; I am a gentleman.

KATHARINA: That I'll try. [She strikes him.]

(Tongue, tail, sting, clitoris -- all tangled together. Who's on first?)

Bruce Deitrick Price explains theories and methods, and intellectual stuff, on his site Improve-Education.org.