The classical dowry, secretary, or beauty contest problem is extended in several directions. In trying to find sequentially the maximum of a random sequence of fixed length, the chooser can have one or several choices (section 2), no information about the distribution of the values (section 2), or at the other extreme, full information about the distribution and the value of the observation itself (section 3). He can have an opponent who alters the properties of the sequence (section 4). The payoff function may be 0 or 1 (sections 2–4), or it may be the value of the observation itself as in certain investment problems (section 5). Both theoretical and numerical results are given for optimum and near optimum play.