Large precision matrix estimation is a crucial problem in high-dimensional multivariate data analytics. When the data exhibit heavy-tailedness or outliers, existing estimators based on Gaussian distribution assumption become inefficient. In this paper, we introduce tLasso, a novel adaptive model for robust precision matrix estimation based on an \ell_{1}-penalized t-distribution log-likelihood function. We assume the degree of freedom parameter of the t-distribution to be unknown a priori, adapting to the arbitrary heavy-tailedness of data distributions. Then we propose a regularized MCECM algorithm to solve the non-convex estimation problem. Theoretically, we prove the non-asymptotic bound on the estimation error, which establishes the computational and statistical guarantees of the algorithm. Numerical simulations validate the effectiveness of the proposed estimator.