An algebra is subassociative if the associator $[x, y, z]$ of any three elements $x, y, z$ is their linear combination. In this paper we prove that any such algebra is Lie-admissible and that almost any such algebra is proper in the sense that there exists an invariant bilinear form $A$ for which there holds the following identity: $[x, y, z] = A(y, z)x - A(x, y)z$, which enables a close connection with associative algebras. We discuss also the improper subassociative algebras.