Many engineering systems contain uncertainties that cannot be described by either deterministic or probabilistic approaches. For example, the geometry, material properties, external actions (loads), and boundary conditions may be imprecise in a practical mechanical or structural system. The uncertainties present may be associated with parameters that are vague, imprecise, or linguistic. A fuzzy meshless local Petrov-Galerkin approach is developed in this work for the analysis of imprecisely defined systems. Starting from the basic concepts of fuzzy sets, fuzzy arithmetic, and fuzzy calculus (differentiation and integration) the various steps of the meshless local Petrov-Galerkin approach involved in the derivation of the equations are redefined using fuzzy concepts. The resulting fuzzy equations are derived using a fuzzy version of Gaussian elimination procedure coupled with truncation. A one-dimensional heat-transfer problem involving conduction, convection, and radiation with vaguely defined thermal properties of the material and boundary conditions and two stress-analysis problems (a beam and a plate with a hole) with vaguely defined material properties and external loads are considered to demonstrate the methodology. A truncation scheme is used to overcome the difficulty of overestimating the width of the response characteristics of the system due to repeated fuzzy arithmetic operations in the solution of the fuzzy equations. A simple sequential linearization scheme is used to handle the nonlinear boundary-value problem associated with the radiation boundary condition in the heat-transfer problem. The present approach represents a unique methodology that enables the handling of certain types of imprecisely known data more realistically compared to the existing procedures.