The sieve tubes of the phloem are enigmatic structures. Their role as channels for the distribution of assimilates was established in the 19th century, but their sensitivity to disturbations has hampered the elucidation of their transport mechanisms and its regulation ever since. Ernst Münch's classical monograph of 1930 is generally regarded as the first coherent theory of phloem transport, but the 'Münchian' pressure flow mechanism had been discussed already before the turn of the century. Münch's impact rather rested on his simple physical models of the phloem that visualized pressure flow in an intuitive way, and we argue that the downscaling of such models to realistic, low-Reynolds-number sizes will boost our understanding of phloem transport in this century just as Münch's models did in the previous one. However, biologically meaningful physical models that could be used to test predictions of the many existing mathematical models would have to be designed in analogy with natural phloem structures. Unfortunately, the study of phloem anatomy seems in decline, and we still lack basic quantitative data required for evaluating the plausibility of our theoretical deductions. In this review, we provide a subjective overview of unresolved problems in angiosperm phloem structure research within a functional context.