The design of artificial catalysts able to compete with the catalytic proficiency of enzymes is an intense subject of research. Non-covalent interactions are thought to be involved in several properties of enzymatic catalysis, notably (i) the confinement of the substrates and the active site within a catalytic pocket, (ii) the creation of a hydrophobic pocket in water, (iii) self-replication properties and (iv) allosteric properties. The origins of the enhanced rates and high catalytic selectivities associated with these properties are still a matter of debate. Stabilisation of the transition state and favourable conformations of the active site and the product(s) are probably part of the answer. We present here artificial catalysts and biomacromolecule hybrid catalysts which constitute good models towards the development of truly competitive artificial enzymes.