Possible explanations for the occurrence of instabilities and oscillations in electrocatalytic reactions are studied for three different kinds of external control of the electrochemical cell: potentiostatic control, fixed-potential control in the case where account must be taken of the cell resistance or an external resistor and galvanostatic control. The relation of our models to existing models and experiments is discussed. In this paper we have investigated simple models of electrocatalytic reactions with the aim of determining the possible origins of instabilities and oscillations under three fundamentally different ways of external control of the electrical state of the electrochemical cell. The conclusions can be summarized as follows. Under truly potentiostatic conditions, i.e. when the interfacial potential can be considered constant irrespective of the current flowing through the cell, the origin of instabilities and oscillations should be purely chemical, i.e. autocatalytic surface chemistry, deviations from ideal Langmuir adsorption, (surface) phase transitions etc. Oscillations cannot be explained by simply referring to the negative slope in the current-voltage characteristic of the reaction. Under potentiostatic conditions a simple negative slope in the J-E curve, owing to passivation for example, cannot give rise to oscillations. Only if a suitable ohmic series resistance, be it only the cell's electrolyte resistance or in addition to this an external series resistor, is present can a simple negative slope give rise to spontaneous oscillations. However, not every negative slope will give rise to oscillations in this way. A negative slope in J-E curve due to competitive adsorption between the electroactive species and a poison will not give oscillations, whatever the value of the external resistance. Potential oscillations under galvanostatic conditions cannot be explained solely by a negative polarization resistance. Potential oscillations can arise if some coupled nonelectrochemical process in the system is assumed to be a potential dependent. We have shown this for a simple model in which a potential-dependent adsorption precedes the electron transfer reaction. This potential-dependent adsorption is capable of hiding the negative slope, but the instability is retained and oscillations occur within a potential interval in the J-E curve with positive slope. This property of the model, and its bifurcation structure in general, are in qualitative agreement with several reports of galvanostatic potential oscillations found experimentally. Because of the simplicity of the models presented in this paper, particularly as these are often reduced to two variables, they can only model simple monoperiodic oscillations. Of course, far more complex oscillatory reponses have been observed experimentally. In principle, one can try to model these by introducing a more complex chemistry or by removing some of the steady-state assumptions that we have introduced. However, these extensions will not significantly change our conclusions about the origin of the oscillations. In this respect, our assumption about the uniform current distribution is a much more serious obstacle. It is not intuitively clear how a nonuniform current distribution (whatever its origin) influences the steady-state stability of an electrochemical cell. It seems that this point needs further eleboration in the future. Finally, the reader will have noticed that we have relied heavily on the mathematical conditions for obtaining oscillations. We have deliberately avoided expressing the different states within a single oscillation into chemical or physical terms. Describing the different events taking place sequentially during a spontaneous oscillation is a difficult and rather tricky business, and it is our experience that such a description is either rather untransparent or, when translated into mathematical terms, will not pass the mathematical test for oscillatory behavior.