There are many reasons under the immigration law that a person may be deemed "inadmissible" to the United States and therefore, a consulate would deny a visa. These reasons include health related grounds - including mental disorders and drug additions, crimal related grounds, security related grounds, foreign policy reasons, public charge reasons - ability for financial support, prior immigration violations, and fraud or misrepresentation.

Below is the language of the law, Immigration and Nationality Act (INA), detailing the grounds of "inadmissibility" - reasons a consulate would deny a visa.

INA Sec. 212(a)
See §309 IIRAIRA for effective date & transition provisions.

Classes of aliens ineligible for visas or admission.--Except as otherwise provided in this Act, aliens who are inadmissible under the following paragraphs are ineligible to receive visas and ineligible to be admitted to the United States:

212(a)(1) Health-related grounds.--

212(a)(1)(A) In general.--Any alien--
212(a)(1)(A)(i) who is determined (in accordance with regulations prescribed by the Secretary of Health and Human Services) to have a communicable disease of public health significance, which shall include infection with the etiologic agent for acquired immune deficiency syndrome,

212(a)(1)(A)(ii)
Effective "with respect to applications for immigrant visas or for adjustment of status filed after September 30, 1996." IIRAIRA §341(c).

except as provided in subparagraph (C), who seeks admission as an immigrant, or who seeks adjustment of status to the status of an alien lawfully admitted for permanent residence, and who has failed to present documentation of having received vaccination against vaccine-preventable diseases, which shall include at least the following diseases: mumps, measles, rubella, polio, tetanus and diphtheria toxoids, pertussis, influenza type B and hepatitis B, and any other vaccinations against vaccine-preventable diseases recommended by the Advisory Committee for Immunization Practices,

212(a)(1)(A)(iii) who is determined (in accordance with regulations prescribed by the Secretary of Health and Human Services in consultation with the Attorney General)--

212(a)(1)(A)(iii)(I) to have a physical or mental disorder and behavior associated with the disorder that may pose, or has posed, a threat to the property, safety, or welfare of the alien or others, or

212(a)(1)(A)(iii)(II) to have had a physical or mental disorder and a history of behavior associated with the disorder, which behavior has posed a threat to the property, safety, or welfare of the alien or others and which behavior is likely to recur or to lead to other harmful behavior, or

212(a)(1)(A)(iv) who is determined (in accordance with regulations prescribed by the Secretary of Health and Human Services) to be a drug abuser or addict, is inadmissible.

212(a)(1)(B) Waiver authorized.--For provision authorizing waiver of certain clauses of subparagraph (A), see subsection (g).

212(a)(1)(C) Exception from immunization requirement for adopted children 10 years of age of younger.--Clause (ii) of subparagraph (A) shall not apply to a child who --

212(a)(1)(C)(i) is 10 years of age or younger,

212(a)(1)(C)(ii) is described in section 101(b)(1)(F), and

212(a)(1)(C)(iii) is seeking an immigrant visa as an immediate relative under section 201(b),

if, prior to the admission of the child, an adoptive parent or prospective adoptive parent of the child, who has sponsored the child for admission as an immediate relative, has executed an affidavit stating that the parent is aware of the provisions of subparagraph (A)(ii) and will ensure that, within 30 days of the child's admission, or at the earliest time that is medically appropriate, the child will receive the vaccinations identified in such subparagraph..

212(a)(2) Criminal and related grounds.--

212(a)(2)(A) Conviction of certain crimes.--
212(a)(2)(A)(i) In general.--Except as provided in clause (ii), any alien convicted of, or who admits having committed, or who admits committing acts which constitute the essential elements of--

212(a)(2)(A)(i)(I) a crime involving moral turpitude (other than a purely political offense) or an attempt or conspiracy to commit such a crime, or

212(a)(2)(A)(i)(II) a violation of (or a conspiracy or attempt to violate) any law or regulation of a State, the United States, or a foreign country relating to a controlled substance (as defined in section 102 of the Controlled Substances Act (21 U.S.C. 802)), is inadmissible.

212(a)(2)(A)(ii) Exception.--Clause (i)(I) shall not apply to an alien who committed only one crime if--

212(a)(2)(A)(ii)(I) the crime was committed when the alien was under 18 years of age, and the crime was committed (and the alien released from any confinement to a prison or correctional institution imposed for the crime) more than 5 years before the date of application for a visa or other documentation and the date of application for admission to the United States, or

212(a)(2)(A)(ii)(II) the maximum penalty possible for the crime of which the alien was convicted (or which the alien admits having committed or of which the acts that the alien admits having committed constituted the essential elements) did not exceed imprisonment for one year and, if the alien was convicted of such crime, the alien was not sentenced to a term of imprisonment in excess of 6 months (regardless of the extent to which the sentence was ultimately executed).

212(a)(2)(B) Multiple criminal convictions.--Any alien convicted of 2 or more offenses (other than purely political offenses), regardless of whether the conviction was in a single trial or whether the offenses arose from a single scheme of misconduct and regardless of whether the offenses involved moral turpitude, for which the aggregate sentences to confinement

The term "actually imposed" was deleted after this word by IIRAIRA §322(a)(2)(B). Change applies to "convictions and sentences entered before, on, or after the date of the enactment of this Act. Subparagraphs (B) and (C) of section 240(c)(3) of the Immigration and Nationality Act, as inserted by section 304(a)(3) of [IIRAIRA], shall apply to proving such convictions."were 5 years or more is inadmissible.

212(a)(2)(C) Controlled substance traffickers.--Any alien who the consular officer or the Attorney General knows or has reason to believe--

212(a)(2)(C)(i) is or has been an illicit trafficker in any controlled substance or in any listed chemical (as defined in section 102 of the Controlled Substances Act (21 U.S.C. 802)), or is or has been a knowing aider, abettor, assister, conspirator, or colluder with others in the illicit trafficking in any such controlled or listed substance or chemical, or endeavored to do so; or

212(a)(2)(C)(ii) is the spouse, son, or daughter of an alien inadmissible under clause (i), has, within the previous 5 years, obtained any financial or other benefit from the illicit activity of that alien, and knew or reasonably should have known that the financial or other benefit was the product of such illicit activity, is inadmissible.