Furfurals are important intermediates in the chemical industry. They are typically produced by homogeneous catalysis in aqueous solutions. However, heterogeneously catalyzed processes would be beneficial in view of the principles of green chemistry: the elimination of homogeneous mineral acids makes the reaction mixtures less corrosive, produces less waste, and facilitates easy separation and recovery of the catalyst. Finding an active and stable water-tolerant solid acid catalyst still poses a challenge for the production of furfural (furan-2-carbaldehyde) and 5-(hydroxymethyl)-2-furaldehyde (HMF). Furfural is produced in the dehydration of xylose, and HMF is formed from glucose and fructose in the presence of an acidic catalyst. Bases are not active in dehydration reaction but do catalyze the isomerization of monosaccharides, which is favorable when using glucose as a raw material. In addition to the desired dehydration of monosaccharides, many undesired side reactions take place, reducing the selectivity and deactivating the catalyst. In addition, the catalyst properties play an important role in the selectivity. In this Review, catalytic conversion approaches are summarized, focusing on the heterogeneously catalyzed formation of furfural. The attractiveness of catalytic concepts is evaluated, keeping in mind productivity, sustainability, and environmental footprint.