Small area estimation techniques typically rely on regression models that use both covariates and random effects to explain variation between the areas. However, such models also depend on strong distributional assumptions, require a formal specification of the random part of the model and do not easily allow for outlier-robust inference. We describe a new approach to small area estimation that is based on modelling quantilelike parameters of the conditional distribution of the target variable given the covariates. This avoids the problems associated with specification of random effects, allowing inter-area differences to be characterised by area-specific M-quantile coefficients. The proposed approach is easily made robust against outlying data values and can be adapted for estimation of a wide range of area-specific parameters, including quantiles of the distribution of the target variable in the different small areas. The differences between M-quantile and random effects models are discussed and the alternative approaches to small area estimation are compared using both simulated and real data.