A cellular counting system based on synthetic gene circuits would enable complex biological programming and be used in many biotechnology applications. Although a variety of synthetic memory circuits have been constructed, basic modules that can be assembled into a counting system are lacking. This thesis focuses on engineering a binary counting module, which can alternate between two states in response to a single repeating input signal. The highly directional large serine bacteriophage integrases were utilised as the basis for the synthetic circuits constructed in this study. Integrases and their protein co-factors, the recombination directionality factor (RDF) can change the orientation of a specific DNA segment flanked by two recombination sites. Integrase alone switches the orientation in one direction, and this directionality is reversed by the addition of its corresponding RDF. The two orientations can be used to turn gene expression on and off, leading to distinct output states which can be thought of as representing a single binary digit (0 and 1) heritably stored in the DNA. In this study, three different serine integrase-based synthetic gene circuits for cellular memory and counting were engineered and characterised. A set-reset latch was first constructed. By expressing ϕC31 integrase and co-expressing integrase with RDF Gp3 from two independent inducible systems, the orientation of the invertible DNA in the set-reset latch was inverted and restored respectively. This device demonstrated that ϕC31 integrase can successfully encode information into plasmid DNA. Next, a state-based latch was constructed, in which the gp3 gene was placed inside the invertible DNA segment to couple its transcriptional regulation to the circuit state. Integrase expression triggered by one input signal resulted in inversion of the invertible DNA, placing the gp3 gene in the correct orientation for transcription. Gp3 expression can then be triggered by another input signal to reverse the directionality of integrase, restoring the DNA back to its original configuration. By optimising the stoichiometry and kinetics of integrase and Gp3 expression, efficient switching of both multi-copy plasmid and single copy chromosomal DNA was achieved. Finally, the state-based latch was developed into a binary counting module by introducing a delay mechanism, in which gp3 transcription was inhibited by a state-based repressor during recombination requiring the absence of Gp3. Placing expression of gp3 under the control of the invertible DNA, allowed a single input signal controlling only integrase expression to switch the module between OFF (0) and ON (1). This is the first integrase-based module that generates different outputs in response to the same input signal and a fundamental step towards building a genetic binary counter with large counting capacity.