Scaling behavior of one-dimensional (1D) and two-dimensional (2D) polymers in dilute solution is discussed with the goal of stimulating experimental work by chemists, physicists, and material scientists in the emerging field of 2D polymers. The arguments are based on renormalization-group theory, which is explained for a general audience. Many ideas and methods successfully applied to 1D polymers are found not to work if one goes to 2D polymers. The role of the various states exhibiting universal behavior is turned upside down. It is expected that solubility will be a serious challenge for 2D polymers. Therefore, given the crucial importance of solutions in characterization and processing, synthetic concepts are proposed that allow the local bending rigidity and the molar mass to be tuned and the long-range interactions to be engineered, all with the goal of preventing the polymer from falling into flat or compact states.