Let C be a smooth k-gonal curve of genus g. We study the number of pencils of degree k on C. In case $ggeqk(k-a)/2$ we state a conjecture based on a discussion on plane models for C. From previous work it is known that if C possesses a large number of pencils then C has a special plane model. From this observation the conjectures are split up in two cases : the existence of some types of plane curves should imply the existence of curves C with a given number of pencils; the non-existence of plane curves should imply the non-existence of curves C with some given large number of pencils. The non-existence part only occurs in the range $k(k-1)/2leqgleqk(k-1)/2] if kgeq7$. In this range we prove the existence part of the conjecture and we also prove some non-existence statements. Those result imply the conjecture in that range for $kleq10$. The cases $kleq6$ are handled separately.