The aim of the author is to establish a general theory of semiclassical orthogonal polynomials. To this end he tries first to determine all the sequences of orthogonal polynomials whose derivatives form quasi- orthogonal sequences. It is shown that this problem is equivalent to the determination of all the sequences of orthogonal polynomials which satisfy certain relations as follows: \[ \Phi (x)P'_{n+1}(x)=\sum^{n+t}_{\nu =n-s}\theta_{n,\nu}P_{\nu}(x), \] where \(\Phi(x)\) and \(\theta_{n,\nu}\) are suitable functions and numbers, respectively. Then he establishes an equation which gives all the linear forms related to a given sequence of orthogonal polynomials. He gives also the conditions under which a sequence, orthogonal with respect to a given linearform, is also orthogonal with respect to an another given linear form. Notice that the author’s definition of “quasi-orthogonality” is somewhat different from what is commonly used.