This thesis is concerned with the growth of microscopic mycelial fungi (Section I), and that of macroscopic fungi, which form specialised hyphal structures such as rhizomorphs (Section II). A growth model is developed in Section I in relation to soil organic matter decomposition, dealing with detailed dynamics of carbon and nitrogen. Substrate with a certain carbon:nitrogen ratio is supplied at a constant rate, broken down and then taken up by fungal mycelium. The nutrients are first stored internally in metabolic pools and then incorporated into structural fungal biomass. Analysis of the overall-steady states of the variables (implicitly from a cubic equation) showed that the conditions for existence had a clear biological interpretation. The 'energy' (in terms of carbon) *invested in* breakdown of substrate should be less than the 'energy' *resulting from* breakdown of substrate, leading to a positive carbon balance. For growth the 'energy' necessary for production of structural fungal biomass and for maintenance should be less than this positive carbon balance in the situation where all substrate is colonised. Under the assumption that nutrient dynamics are much faster than the dynamics of fungal biomass and substrate, a quasi-steady analysis was performed. From the resulting simplified model an explicit fungal invasion criterion was derived, which was not possible in the analysis of the original fungal growth model. The fungal invasion criterion takes two forms: one for systems where carbon is limiting, another for systems where nitrogen is limiting. For cases where only carbon is limiting, nitrogen dynamics were excluded from the model, and this further simplification resulted in a model that was fitted to data on growth of the soil-borne plant pathogen *Rhizoctonia solani* . Fungal growth and colonisation of discrete nutrient sites in Petri plates were assessed microscopically for two carbon concentrations of the substrate. Colonisation was faster at the higher carbon concentration. The model predicted a lower asymptote for non-colonised substrate and this value was estimated from the data by non-linear regression for each carbon concentration. A key composite parameter, the positive carbon balance per carbon unit of colonised substrate, was lower for the higher carbon concentration. The carbon decomposition rate was estimated by least squares minimisation, after correction for a lag phase expected after robust handling of the inoculated fungus. The delay in subsequent fungal growth after inoculation was extended when there was less carbon available for physical recovery and physiological adaptation to the new environment. The simplified mean-field model with parameters estimated as described above produced a good fit to the data.

In Section II quantitative studies on the epidemiology of *Armillaria* root rot are reviewed. This fungus is a serious disease in many forests and horticultural tree crops world-wide, and consequently there is much interest in options for avoiding or restricting the spread of disease through growth of the specialised rhizomorphs in soil. Two rhizomorph networks of *A. lutea* growing through a natural soil were observed over areas of 25 m ^{2}in *Pinus nigra* and *Picea abies* tree plantations. Both rhizomorph systems had numerous branches and anastomoses resulting in cyclic paths, i.e. regions of the system that start and end at the same point. Each rhizomorph network exhibited both exploitative and explorative characteristics within its overall network structure. One of the observed rhizomorph networks of *A. lutea* was restricted to the cyclic paths only, and the resulting graph was drawn in the plane. The plane graph consisted of 169 rhizomorphs, termed edges, and 107 rhizomorph nodes, termed vertices. The connectivity of the rhizomorph network was explored by focusing on each bridge, i.e. an edge whose removal disconnects the graph into two components. In only two instances was a nutrient source connected to the cycles, and disruption of these two connecting edges would remove the whole network from the sources. A shortest path from a given vertex to a nutrient source was defined in terms of number of edges, and also in terms of length (m). The length of the edges enclosing the faces, i.e. two-dimensional regions defined by the edges in the plane drawing, showed that the fungus exhibited both exploitative and explorative growth, and we speculate about the underlying reasons for these foraging strategies. The introduction of graph-theoretic concepts to fungal growth might lead to an improved ecological understanding of fungal networks in general, provided that relevant biological interpretations can be made.