An almost Pythagorean triple (APT) is a solution in integers of the equation \(x^ 2+y^ 2=z^ 2+1\). A nearly Pythagorean triple (NPT) is a solution in integers of the equation \(x^ 2+y^ 2=z^ 2-1\). All APT’s and an infinite number of NPT’s are found. It is shown that each APT gives an NPT and vice-versa. A complete solution of a more general equation than the one associated with NPT’s was discovered by Catalan in 1885 [see A. B. Ayoub, ibid. 57, 222-223 (1984; Zbl 0549.10009)].