We deal with the existence of positive solutions for the following fractional Schrödinger equation: \varepsilon ^{2s} (-\Delta)^{s} u + V(x) u = f(u) \quad \text{in } \mathbb{R}^{N}, where \varepsilon>0 is a parameter, s\in (0, 1) , N\geq 2 , (-\Delta)^{s} is the fractional Laplacian operator, and V\colon \mathbb{R}^{N}\rightarrow \mathbb{R} is a positive continuous function. Under the assumptions that the nonlinearity f is either asymptotically linear or superlinear at infinity, we prove the existence of a family of positive solutions which concentrates at a local minimum of V as \varepsilon tends to zero.