We propose a direction it the field of statistics which we will call functional object analysis. This subfields considers the analysis of functional objects defined on continuous domains. In this setting we will focus on model-based statistics, with a particularly emphasis on mixed-effect formulations, where the observed functional signal is assumed to consist of both fixed and random functional effects. This thesis takes the initial steps toward the development of likelihood-based methodology for functional objects. We first consider analysis of functional data defined on high-dimensional Euclidean spaces under the effect of additive spatially correlated effects, and then move on to consider how to include data alignment in the statistical model as a nonlinear effect under additive correlated noise. In both cases, we will give directions on how to generalize the methodology to more complex data setups. Finally, we consider various extensions and future directions.