Interactions of shock waves with gas-gas and gas-liquid interfaces (under both slow-fast and fast-slow configurations) are studied using the recently developed Adaptive Characteristics-based Matching (aCBM) method for capturing interfaces in compressible multi-fluid media. First, we verify our approach for the gas-gas case; a class of problems for which a substantial body of knowledge already exists. Then, we consider slow-fast, gas-liquid interfaces under weak shocks, and fast-slow, liquid-gas interfaces under strong shocks. The very high acoustic impedance mismatch situation here creates significant numerical (simulation) and experimental (visualization) difficulties, and the literature for it is meager and sporadic. Compared to gas-gas interfaces we note both similarities and differences. We discuss the sources for these differences, as well as potential implications of generalizing and embedding such results in multi-dimensional simulation schemes towards improving their front-capturing performance. (c) 2005 Elsevier Ltd. All rights reserved.