Recent advances in Large Language Models (LLMs) have propelled conversational AI from traditional dialogue systems into sophisticated agents capable of autonomous actions, contextual awareness, and multi-turn interactions with users. Yet, fundamental questions about their capabilities, limitations, and paths forward remain open. This survey paper presents a desideratum for next-generation Conversational Agents - what has been achieved, what challenges persist, and what must be done for more scalable systems that approach human-level intelligence. To that end, we systematically analyze LLM-driven Conversational Agents by organizing their capabilities into three primary dimensions: (i) Reasoning - logical, systematic thinking inspired by human intelligence for decision making, (ii) Monitor - encompassing self-awareness and user interaction monitoring, and (iii) Control - focusing on tool utilization and policy following. Building upon this, we introduce a novel taxonomy by classifying recent work on Conversational Agents around our proposed desideratum. We identify critical research gaps and outline key directions, including realistic evaluations, long-term multi-turn reasoning skills, self-evolution capabilities, collaborative and multi-agent task completion, personalization, and proactivity. This work aims to provide a structured foundation, highlight existing limitations, and offer insights into potential future research directions for Conversational Agents, ultimately advancing progress toward Artificial General Intelligence (AGI). We maintain a curated repository of papers at: https://github.com/emrecanacikgoz/awesome-conversational-agents.