Allosteric RNAs operate as molecular switches that alter folding and function in response to ligand binding. A common type of natural allosteric RNAs is the riboswitch; designer RNAs with similar properties can be created by RNA engineering. We describe a computational approach for designing allosteric ribozymes triggered by binding oligonucleotides. Four universal types of RNA switches possessing AND, OR, YES and NOT Boolean logic functions were created in modular form, which allows ligand specificity to be changed without altering the catalytic core of the ribozyme. All computationally designed allosteric ribozymes were synthesized and experimentally tested in vitro. Engineered ribozymes exhibit >1,000-fold activation, demonstrate precise ligand specificity and function in molecular circuits in which the self-cleavage product of one RNA triggers the action of a second. This engineering approach provides a rapid and inexpensive way to create allosteric RNAs for constructing complex molecular circuits, nucleic acid detection systems and gene control elements.