This dissertation presents the steps employed to activate and utilise analogue memristive devices in conventional analogue circuits and beyond.
TiO2 memristors are mainly utilised in this study, and their large variability in operation in between similar devices is identified.
A specialised memristor characterisation instrument is designed and built to mitigate this issue and to allow access to large numbers of devices at a time.
Its performance is quantified against linear resistors, crossbars of linear resistors, stand-alone memristive elements and crossbars of memristors.
This platform allows for a wide range of different pulsing algorithms to be applied on individual devices, or on crossbars of memristive elements, and is used throughout this dissertation.

Different ways of achieving analogue resistive switching from any device state are presented.
Results of these are used to devise a state-of-art biasing parameter finder which automatically extracts pulsing parameters that induce repeatable analogue resistive switching.
IV measurements taken during analogue resistive switching are then utilised to model the internal atomic structure of two devices, via fittings by the Simmons tunnelling barrier model.
These reveal that voltage pulses modulate a nano-tunnelling gap along a conical shape.

Further retention measurements are performed which reveal that under certain conditions, TiO2 memristors become volatile at short time scales.
This volatile behaviour is then implemented into a novel SPICE volatile memristor model.

These characterisation methods of solid-state devices allowed for inclusion of TiO2 memristors in practical electronic circuits.
Firstly, in the context of large analogue resistive crossbars, a crosspoint reading method is analysed and improved via a 3-step technique.
Its scaling performance is then quantified via SPICE simulations.
Next, the observed volatile dynamics of memristors are exploited in two separate sequence detectors, with applications in neuromorphic engineering.
Finally, the memristor as a programmable resistive weight is exploited to synthesise a memristive programmable gain amplifier and a practical memristive automatic gain control circuit.