This paper establishes the asymptotic distribution of an extremum estimator when the true parameter lies on the boundary of the parameter space. The boundary may be linear, curved, and/or kinked. Typically the asymptotic distribution is a function of a multivariate normal distribution in models without stochastic trends and a function of a multivariate Brownian motion in models with stochastic trends. The results apply to a wide variety of estimators and models. Examples treated in the paper are: (i) quasi‐ML estimation of a random coefficients regression model with some coefficient variances equal to zero and (ii) LS estimation of an augmented Dickey‐Fuller regression with unit root and time trend parameters on the boundary of the parameter space.