Small area estimation has long been a popular and important research topic in survey statistics. For the basic area level model, popularly known as Fay-Herriot model, we first make inference without any distributional assumptions with the exception of a few moment assumptions. In the process, we propose a new method of model parameter estimation, study its statistical properties and use the resulting parameter estimators as components in small area estimators. The second order approximation of the mean squared error of the proposed small area estimators is derived, and we also describe a second order correct estimator of the mean squared error. Then we develop confidence intervals for the small area parameters that are second order correct under normal distributional assumptions. For the unit level model, popularly known as nested-error regression model, we introduce a model-based design consistent estimator for a finite population domain mean.