This thesis studies physical and stochastic aspects of microorganisms. From the point of view of $physics$, the studies in this thesis are motivated by the goal of gaining biological insight using the machinery of physics and mathematics. From the point of view of $biology$, the studies in this thesis focus primarily on choanoflagellates, eukaryotes that are the closest living unicellular relatives of animals. This choice of model organism was motivated by the important biological question of the origin of multicellularity. Why was it that single-celled organisms evolved to become multicellular? In particular, we study closely the species $Salpingoeca\; rosetta$, which has the ability to form colonies that resemble true multicellular organisms.

A large part of this thesis deals with the random walks of microorganisms. We study these active random walks both for single cells and those composed of individual organisms adhered together. The latter colonial random walkers are typified by choanoflagellates. We develop quantitative theories and use these to extract physical parameters.

The increasing ocean oxygen levels in the Precambrian era are thought to be an important factor in the emergence of complex multicellular, animal life. As a first step, we address this situation by studying the response of $S.\; rosetta$ to oxygen gradients. We find that $S.\; rosetta$ displays positive aerotaxis. Analysis of the spatial population distributions provides evidence for logarithmic sensing of oxygen, which enhances sensing in low oxygen neighbourhoods. Analysis of search strategy models on the experimental colony trajectories finds that choanoflagellate aerotaxis is consistent with stochastic navigation, the statistics of which are captured using an effective continuous version of classical run-and-tumble chemotaxis.

We compare this continuous run-to-tumble with the run-and-tumble seen in bacteria by formulating a general model for persistent run-and-tumble. We find that although an optimal persistence does exist for a given tumble frequency, in the full parameter space there is a continuum of optimal solutions. We develop this model further by introducing finite tumble times.

Efficient uptake of prey and nutrients from the environment is an important component in the fitness of all microorganisms, and its dependence on size may reveal clues to the origins of evolutionary transitions to multicellularity. We examine these issues in depth for choanoflagellates, finding that in the absence of other requirements and in a homogeneously nutritious environment, the optimal strategy to maximise filter feeding is to swim fast which favours swimming unicells. In contrast, in large external flows, a sessile form becomes advantageous. Effects of prey diffusion are discussed and are also found to be advantageous for the swimming unicell.

Finally, we consider the switching between synchronous and anti-synchronous beating of flagella in the green alga $Chlamydomonas$, a phenomenon that results in run-and-tumble behaviour in eukaryotes. We develop a theoretical model to describe this beating and use it to argue that the synchrony itself is obtained intracellularly, whereas the flagella shapes are most likely strongly influenced by hydrodynamic interactions.