Suppose that μ is a Radon measure on Rd, which may be nondoubling. The only condition on μ is the growth condition, namely, there is a constant C0 > 0 such that for all x ∈ supp (μ) and r > 0,μ(B(x, r)) ≤ C0rn,where 0 < n ≤ d. In this paper, the authors establish a theory of Besov spaces Bspq(μ) for 1 ≤ p, q ≤ ∞ and |s| < θ, where θ > 0 is a real number which depends on the non-doubling measure μ, C0, n and d. The method used to define these spaces is new even for the classical case. As applications, the lifting properties of these spaces by using the Riesz potential operators and the dual spaces are obtained.