How to deal with criticism?

How should one properly handle cases of insults, criticism and discrimination, according to the Buddhist teachings?

Here's what the Buddha had to say to the monks:

For even this external earth element, great as it is, can be seen as impermanent, can be seen as liable to destruction, liable to fall, liable to change; so what of this body, which is clung-to by craving and lasts but a while? There can be no (considering) that as ‘I’ or ‘mine’ or ‘I am.’

So then, (thus having seen this element as it actually is) if others abuse and scold and curse and threaten a bhikkhu, he understands thus: ‘This painful feeling born of ear-contact has arisen in me. That is dependent, not independent. Dependent on what? Dependent on contact.’ Then he sees that contact is impermanent, that feeling is impermanent, that perception is impermanent, that mental formations are impermanent, and that consciousness is impermanent. And his mind enters into that very object (taking it just as an impersonal) element, and acquires confidence, steadiness and decision (herein).
_{-MN 28, The Greater Discourse on the Elephant-footprint Simile}

"Monks, even if bandits were to carve you up savagely, limb by limb, with a two-handled saw, he among you who let his heart get angered even at that would not be doing my bidding. Even then you should train yourselves: 'Our minds will be unaffected and we will say no evil words. We will remain sympathetic, with a mind of good will, and with no inner hate. We will keep pervading these people with an awareness imbued with good will and, beginning with them, we will keep pervading the all-encompassing world with an awareness imbued with good will — abundant, expansive, immeasurable, free from hostility, free from ill will.' That's how you should train yourselves.

"Monks, if you attend constantly to this admonition on the simile of the saw, do you see any aspects of speech, slight or gross, that you could not endure?"

"No, lord."

"Then attend constantly to this admonition on the simile of the saw. That will be for your long-term welfare & happiness."
_{-MN 21, The Simile of the Saw}

"Punna, the Sunaparanta people are fierce. They are rough. If they insult and ridicule you, what will you think?"

"If they insult and ridicule me, I will think, 'These Sunaparanta people are civilized, very civilized, in that they don't hit me with their hands.' That is what I will think, O Blessed One. That is what I will think, O One Well-gone."

"But if they hit you with their hands, what will you think?"

"...I will think, 'These Sunaparanta people are civilized, very civilized, in that they don't hit me with a clod.'..."

"But if they hit you with a clod...?"

"...I will think, 'These Sunaparanta people are civilized, very civilized, in that they don't hit me with a stick.'..."

"But if they hit you with a stick...?"

"...I will think, 'These Sunaparanta people are civilized, very civilized, in that they don't hit me with a knife.'..."

"But if they hit you with a knife...?"

"...I will think, 'These Sunaparanta people are civilized, very civilized, in that they don't take my life with a sharp knife.'..."

"But if they take your life with a sharp knife...?"

"If they take my life with a sharp knife, I will think, 'There are disciples of the Blessed One who — horrified, humiliated, and disgusted by the body and by life — have sought for an assassin, but here I have met my assassin without searching for him.' That is what I will think, O Blessed One. That is what I will think, O One Well-gone."

"Good, Punna, very good. Possessing such calm and self-control you are fit to dwell among the Sunaparantans. Now it is time to do as you see fit."
_{-SN 35.88, To Punna}

In addition, with cases of blame or disrepute, you might want to take a look at this answer.