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The Equal Protection Clause of the Fourteenth Amendment to the US Constitution provides as follows:
No State shall {...} deny to any person within its jurisdiction the equal protection of the laws.
As it is currently understood, the Equal Protection Clause forbids invidious disparate treatment of similarly situated individuals (the Due Process Clause, not the Equal Protection Clause, makes the Bill of Rights enforceable as against the states). Contrary to the assertion made above by AngelKnight2780, the Due Process Clause is the only basis in the Constitution for incorporation of the Bill of Rights. See e.g. Hurtado v. California, 110 U.S. 516 (1884) (relying on Due Process Clause), Palko v. Connecticut, 302 U.S. 319 (1937) (deciding whether Due Process Clause incorporated Double Jeopardy Clause), Duncan v. Louisiana, 391 U.S. 145, 147 (1968) (evaluating incorporation of criminal procedure guarantees under Due Process Clause).

In evaluating Equal Protection Claims, courts first look to determine what sort of classification, if any, is created by a challenged enactment or policy. There are three basic types of classifications: suspect classifications (those based on race, national origin, religion, or alienage (those classifications that Congress specifically sought to abolish when enacting the Fourteenth Amendment), quasi-suspect classifications (gender and other similar classifications), and everything else. Courts will additionally look to whether the classification burdens the exercise of a fundamental right.

Having determined what sort of classification has been created, courts will apply differing levels of scrutiny to the challenged classification:

If the challenged statute or policy classifies individuals based on race, religion, national origin, or alienage and/or burdens the exercise of a fundamental right (a right "implicit in the concept of ordered liberty"), courts will apply strict scrutiny. Under strict scrutiny, courts will only uphold the policy or statute if it is "narrowly tailored" to serve a "compelling state interest" and no other means of serving that interest are available that do not include the suspect classification.(See also standard of review)

If the challenged statute or policy classifies based on gender, a lesser degree of scrutiny applies, referred to as "intermediate" or "heightened" scrutiny. In this case, an "exceedingly persuasive justification is required," which may not rely on generalisations about the abilities, preferences, etc. of one sex or another.

Both of these tests tend generally to invalidate statutes.

For the rabble of non-suspect classifications that don't burden fundamental rights (e.g. a statute requiring people who talk on cell phones in public in annoyingly loud voices to pay an "annoyance tax" for using public buildings), there is the "rational basis" test. This is the lowest standard available, although recent cases have shown that it isn't always a rubber stamp. Basically, if there is a "rational basis" for enacting the statute or a policy, any rational basis at all, it must be upheld.