Consider the problem of testing Z ~ ℙᵐ vs Z ~ ℚᵐ from m samples. Generally, to achieve a small error rate it is necessary and sufficient to have m≍1/ϵ², where ϵ measures the separation between ℙ and ℚ in total variation (𝖳𝖵). Achieving this, however, requires complete knowledge of the distributions ℙ and ℚ and can be done, for example, using the Neyman-Pearson test. In this paper we consider a variation of the problem, which we call likelihood-free (or simulation-based) hypothesis testing, where access to ℙ and ℚ (which are a priori only known to belong to a large non-parametric family P) is given through n iid samples from each. We demostrate existence of a fundamental trade-off between n and m given by nm ≍ n²_𝖦𝗈𝖥(ϵ,P), where n_𝖦𝗈𝖥 is the minimax sample complexity of testing between the hypotheses H₀: ℙ=ℚ vs H₁: 𝖳𝖵(ℙ,ℚ)≥ϵ. We show this for three non-parametric families P: β-smooth densities over [0,1]d, the Gaussian sequence model over a Sobolev ellipsoid, and the collection of distributions P on a large alphabet [k] with pmfs bounded by c/k for fixed c. The test that we propose (based on the L²-distance statistic of Ingster) simultaneously achieves all points on the tradeoff curve for these families. In particular, when m≫1/ϵ² our test requires the number of simulation samples n to be orders of magnitude smaller than what is needed for density estimation with accuracy ≍ϵ (under 𝖳𝖵). This demonstrates the possibility of testing without fully estimating the distributions.