At common law, admissions were admissible. A statement could only be excluded by a showing of involuntariness, unfairness, or that the circumstances under which the statement was obtained was improper or illegal.
In the United States, "Admission by a party-opponent" is explicitly excepted from hearsay under the Federal Rules of Evidence. Rule 801(d)(2). Among several types of admissions, the rule notes that an admission can be the "party's own statement" or a statement in which the "party has manifested an adoption or belief in its truth."