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Deep learning models have achieved high performance across different domains, such as medical decision-making, autonomous vehicles, decision support systems, among many others. However, despite this success, the inner mechanisms of these models are opaque because their internal representations are too complex for a human to understand. This opacity makes it hard to understand the how or the why of the predictions of deep learning models. There has been a growing interest in model-agnostic methods that make deep learning models more transparent and explainable to humans. Some researchers recently argued that for a machine to achieve human-level explainability, this machine needs to provide human causally understandable explanations, also known as causability. A specific class of algorithms that have the potential to provide causability are counterfactuals. This paper presents an in-depth systematic review of the diverse existing literature on counterfactuals and causability for explainable artificial intelligence (AI). We performed a Latent Dirichlet topic modelling analysis (LDA) under a Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) framework to find the most relevant literature articles. This analysis yielded a novel taxonomy that considers the grounding theories of the surveyed algorithms, together with their underlying properties and applications to real-world data. Our research suggests that current model-agnostic counterfactual algorithms for explainable AI are not grounded on a causal theoretical formalism and, consequently, cannot promote causability to a human decision-maker. Furthermore, our findings suggest that the explanations derived from popular algorithms in the literature provide spurious correlations rather than cause/effects relationships, leading to sub-optimal, erroneous, or even biased explanations. Thus, this paper also advances the literature with new directions and challenges on promoting causability in model-agnostic approaches for explainable AI. • A survey on model-agnostic counterfactual approaches for XAI. • A novel taxonomy for model-agnostic counterfactual approaches for XAI. • A set of properties for causability systems for XAI. • Opportunities and challenges for causability systems based on modelagnostic counterfactual approaches. [ABSTRACT FROM AUTHOR]