Physical-layer security is emerging as a promising paradigm of securing wireless communications against eavesdropping between legitimate users, when the main link spanning from source to destination has better propagation conditions than the wiretap link from source to eavesdropper. In this paper, we identify and analyze the tradeoffs between the security and reliability of wireless communications in the presence of eavesdropping attacks. Typically, the reliability of the main link can be improved by increasing the source's transmit power (or decreasing its date rate) to reduce the outage probability, which unfortunately increases the risk that an eavesdropper succeeds in intercepting the source message through the wiretap link, since the outage probability of the wiretap link also decreases when a higher transmit power (or lower date rate) is used. We characterize the security-reliability tradeoffs (SRT) of conventional direct transmission from source to destination in the presence of an eavesdropper, where the security and reliability are quantified in terms of the intercept probability by an eavesdropper and the outage probability experienced at the destination, respectively. In order to improve the SRT, we then propose opportunistic relay selection (ORS) and quantify the attainable SRT improvement upon increasing the number of relays. It is shown that given the maximum tolerable intercept probability, the outage probability of our ORS scheme approaches zero for $N \to \infty$, where $N$ is the number of relays. Conversely, given the maximum tolerable outage probability, the intercept probability of our ORS scheme tends to zero for $N \to \infty$.