  In this thesis we present few theoretical studies of the models of
self-organized criticality. Following a brief introduction of self-organized
criticality, we discuss three main problems. The first problem is about growing
patterns formed in the abelian sandpile model (ASM). The patterns exhibit
proportionate growth where different parts of the pattern grow in same rate,
keeping the overall shape unchanged. This non-trivial property, often found in
biological growth, has received increasing attention in recent years. In this
thesis, we present a mathematical characterization of a large class of such
patterns in terms of discrete holomorphic functions. In the second problem, we
discuss a well known model of self-organized criticality introduced by Zhang in
1989. We present an exact analysis of the model and quantitatively explain an
intriguing property known as the emergence of quasi-units. In the third
problem, we introduce an operator algebra to determine the steady state of a
class of stochastic sandpile models.
