I offered the following answer to the titular question at Quora:
Section 8.01(a) of the Model Business Corporation Act (in force in over 2/3 of the states) provides that: “Except as may be provided in an agreement authorized under section 7.32, each corporation shall have a board of directors.”
In turn, section 7.32(a) provides that: “An agreement among the shareholders of a corporation that complies with this section is effective among the shareholders and the corporation even though it is inconsistent with one or more other provisions of this Act in that it: (1) eliminates the board of directors ….”
Note, however, that such an agreement must be unanimous and be made known to the corporation. It must be disclosed on share certificates.
If the agreement eliminates the board of directors, there is no prescribed replacement. As the Official Comment to section 8.01(a) states, “the shareholders of a corporation may, in an agreement that satisfies the requirements of section 7.32, dispense with a board of directors and structure the corporation’s management and governance to address specific needs of the enterprise.”
Although section 7.32 is not limited to closely held corporations, many states have limited the option of eliminating the bard of directors to closely held corporations.
Indeed, it is essentially unheard of for publicly held corporations to have such agreements and equally unheard of for publicly held corporations to eliminate the board. See, e.g., POST PROPERTIES, INC., SEC No Action Ltrs. WSB File No. 0329200417 (“It is inconceivable, for example, that shareholders of a publicly-traded company such as Post Properties would attempt to eliminate the board of directors entirely, to designate an outside person to manage its business, or to give to one of its shareholders the power to cause corporate dissolution. These limits are totally incompatible with public ownership ….”
Delaware General Corporation Law section 141(a) provides that: “The business and affairs of every corporation organized under this chapter shall be managed by or under the direction of a board of directors, except as may be otherwise provided in this chapter or in its certificate of incorporation. If any such provision is made in the certificate of incorporation, the powers and duties conferred or imposed upon the board of directors by this chapter shall be exercised or performed to such extent and by such person or persons as shall be provided in the certificate of incorporation.”
