We calculate the photoemission spectrum in a cold Fermi gas measured by the JILA group (Debbie Jin). Such experiments gives direct information about the single-particle properties of the fermions in the BCS-BEC crossover region. By including pairing fluctuations within a T -matrix approximation, as well as effects of a harmonic trap within the local density approximation, we show that the spatially inhomogeneous pair fluctuations ( molecules) produced by the trap potential is crucial to understand the observed photoemission spectrum. In the crossover region, while strong pair fluctuations lead to the so-called pseudogap phenomenon in the trap center, such strong-coupling effects are small around the edge of the gas. Our theoretical results agree well with the latest photoemission data by JILA group reported this year. I will also briefly talk about our results on the stability of a superfluid Fermi gas in a moving optical lattice.