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In this talk, we deal with the problem of embedding some continuous-time processes of unit length into a Brownian path. Beginning with a discrete version of the problem, we derive the asymptotics of the expected waiting time for several interesting patterns. These suggest corresponding results on the existence or non-existence of continuous paths embedded in Brownian motion. With further effort, we are able to prove some of these existence and non-existence results by various stochastic analysis tools: potential theory, filling scheme, Palm theory...etc. This is based on joint work with Jim Pitman.