This paper explores the use of reconfigurable intelligent surfaces (RIS) in mitigating cross-system interference in spectrum sharing and secure wireless applications. Unlike conventional RIS that can only adjust the phase of the incoming signal and essentially reflect all impinging energy, or active RIS, which also amplify the reflected signal at the cost of significantly higher complexity, noise, and power consumption, an absorptive RIS (ARIS) is considered. An ARIS can in principle modify both the phase and modulus of the impinging signal by absorbing a portion of the signal energy, providing a compromise between its conventional and active counterparts in terms of complexity, power consumption, and degrees of freedom (DoFs). We first use a toy example to illustrate the benefit of ARIS, and then we consider three applications: (1) Spectral coexistence of radar and communication systems, where a convex optimization problem is formulated to minimize the Frobenius norm of the channel matrix from the communication base station to the radar receiver; (2) Spectrum sharing in device-to-device (D2D) communications, where a max-min scheme that maximizes the worst-case signal-to-interference-plus-noise ratio (SINR) among the D2D links is developed and then solved via fractional programming; (3) The physical layer security of a downlink communication system, where the secrecy rate is maximized and the resulting nonconvex problem is solved by a fractional programming algorithm together with a sequential convex relaxation procedure. Numerical results are then presented to show the significant benefit of ARIS in these applications.