We develop a systematic way for constructing bispectral algebras of commuting ordinary differential operators of any rank $N$. It combines and unifies the ideas of Duistermaat-Grünbaum and Wilson. Our construction is completely algorithmic and enables us to obtain all previously known classes or individual examples of bispectral operators. The method also provides new broad families of bispectral algebras which may help to penetrate deeper into the problem.

46 pages, LaTeX2e, uses amsfonts.sty and latexsym.sty, no figures, rearrangement of the introduction, skipping Conjecture 0.2 of the first version, to appear in Communications in Mathematical Physics