Let \(M_ n\) be the set of all complex \(n\times n\) matrices and \(A\in M_ n\). The authors discuss the problem of triangularizing A by complex orthogonal similarity and consimilarity, i.e. factorizing \(A=Q\Delta Q^ T\) or \(A=Q\Delta Q^*\), where \(Q\in M_ n\) is complex orthogonal and \(\Delta \in M_ n\) upper triangular. It is proved that A can be triangularized by orthogonal similarity under one of the following conditions: a) \(Z^{-1}AZ\) is upper triangular (diagonal) for a nonsingular matrix \(Z=[z_ 1,...,z_ n]\in M_ n\) such that \(\det ([z_ 1,...,z_ i]^ T\cdot [z_ 1,...,z_ i])\neq 0\) for \(i=1,...,n\) (Theorem 1 and Corollary 3); b) \(A\bar A\) is real, \(A+\bar A\) has only real, and \(A-\bar A\) only imaginary eigenvalues (L. 4). A nonzero vector \(x\in C^ n\) such that \(A\bar x=\lambda x\) is said to be a coneigenvector of A. Set \(F(A)=\{x^*Ax: x^*x=1,\quad x\in C^ n\}.\) If \(0\not\in F(A)\) and \(A\bar A\) has a nonnegative eigenvalue, then A has a nonisotropic coneigenvector x \((x^ Tx\neq 0)\) (L. 8). Suppose that, whenever \(Q\in M_ n\) is complex orthogonal and \(QAQ^*\) some coneigenvector of A is nonisotropic. Then \(QAQ^*\) is upper triangular for some complex orthogonal \(Q\in M_ n\) (Theorem 10). Let A be such that \(0\not\in F(A)\). There exists a complex orthogonal matrix \(Q\in M_ n\) such that \(QAQ^*\) is upper triangular iff all eigenvalues of \(A\bar A\) are nonnegative (Corollary 11). A is positive definite iff there exists a complex orthogonal matrix \(Q\in M_ n\) such that \(QAQ^*\) is diagonal with positive main diagonal elements (Corollary 13).